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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linear dynamical system</span></span>
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<p><b>Linear dynamical systems</b> are <a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a> whose <a href="Evolution_function" class="mw-redirect" title="Evolution function">evolution functions</a> are <a href="Linear" class="mw-redirect" title="Linear">linear</a>. While dynamical systems, in general, do not have <a href="Closed-form_expression" title="Closed-form expression">closed-form solutions</a>, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the <a href="Equilibrium_points" class="mw-redirect" title="Equilibrium points">equilibrium points</a> of the system and approximating it as a linear system around each such point.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2></div>
<p>In a linear dynamical system, the variation of a state vector
(an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span>-dimensional <a href="Vector_space" title="Vector space">vector</a> denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi mathvariant="bold">x</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>
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</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span>) equals a constant matrix
(denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>
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</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span>) multiplied by
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>
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</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span>. This variation can take two forms: either
as a <a href="Flow_(mathematics)" title="Flow (mathematics)">flow</a>, in which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>
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</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> varies
continuously with time
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {x} (t)}">
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<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {x} (t)}</annotation>
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</math></span><img src="./fae1082637d5eb9e96839c77cf34307efd8ec312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.129ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {x} (t)}" loading="lazy"></span></dd></dl>
<p>or as a mapping, in which
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>
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</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> varies in <a href="Discrete_time" class="mw-redirect" title="Discrete time">discrete</a> steps
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{m+1}=\mathbf {A} \mathbf {x} _{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mn>1</mn>
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
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<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
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</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{m+1}=\mathbf {A} \mathbf {x} _{m}}</annotation>
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</math></span><img src="./ec57e63c5f83211b3ab76f47f534e1ea7a7e8e37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.39ex; height:2.509ex;" alt="{\displaystyle \mathbf {x} _{m+1}=\mathbf {A} \mathbf {x} _{m}}" loading="lazy"></span></dd></dl>
<p>These equations are linear in the following sense: if
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t)}</annotation>
</semantics>
</math></span><img src="./385450fed781e07652f160faa271a7cc17fdcbc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.06ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} (t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} (t)}</annotation>
</semantics>
</math></span><img src="./1a85cc905ba3c42b0f18262a7073b5738df7337a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.06ex; height:2.843ex;" alt="{\displaystyle \mathbf {y} (t)}" loading="lazy"></span>
are two valid solutions, then so is any <a href="Linear_combination" title="Linear combination">linear combination</a>
of the two solutions, e.g.,
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} (t)\ {\stackrel {\mathrm {def} }{=}}\ \alpha \mathbf {x} (t)+\beta \mathbf {y} (t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-REL">
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo>=</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
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<mi mathvariant="normal">f</mi>
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</mrow>
<mtext> </mtext>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
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<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} (t)\ {\stackrel {\mathrm {def} }{=}}\ \alpha \mathbf {x} (t)+\beta \mathbf {y} (t)}</annotation>
</semantics>
</math></span><img src="./f5b7a4f216faeb7dfdc39826332de5cf4982c7e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.034ex; height:3.843ex;" alt="{\displaystyle \mathbf {z} (t)\ {\stackrel {\mathrm {def} }{=}}\ \alpha \mathbf {x} (t)+\beta \mathbf {y} (t)}" loading="lazy"></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>β<!-- β --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>
</semantics>
</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span>
are any two <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>. The matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>
</semantics>
</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span>
need not be <a href="Symmetry_in_mathematics#Symmetry_in_linear_algebra" title="Symmetry in mathematics">symmetric</a>.
</p><p>Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a>. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.
</p>
<div class="mw-heading mw-heading2"><h2 id="Solution_of_linear_dynamical_systems">Solution of linear dynamical systems</h2></div>
<p>If the initial vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{0}\ {\stackrel {\mathrm {def} }{=}}\ \mathbf {x} (t=0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<msub>
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<mi mathvariant="bold">x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-REL">
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mo>=</mo>
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<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
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<mi>t</mi>
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<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{0}\ {\stackrel {\mathrm {def} }{=}}\ \mathbf {x} (t=0)}</annotation>
</semantics>
</math></span><img src="./6324e3c265b52d1efea40733f144959c5b57cd9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.203ex; height:3.843ex;" alt="{\displaystyle \mathbf {x} _{0}\ {\stackrel {\mathrm {def} }{=}}\ \mathbf {x} (t=0)}" loading="lazy"></span>
is aligned with a <a href="Right_eigenvector" class="mw-redirect" title="Right eigenvector">right eigenvector</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{k}}</annotation>
</semantics>
</math></span><img src="./01da6ca6fcfe0ac44e859a2bad84c3d90846f195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.191ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{k}}" loading="lazy"></span> of
the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>
</semantics>
</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span>, the dynamics are simple
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {r} _{k}=\lambda _{k}\mathbf {r} _{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {r} _{k}=\lambda _{k}\mathbf {r} _{k}}</annotation>
</semantics>
</math></span><img src="./d9fba183976b06b86d43966928e97160f7e11b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.993ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {r} _{k}=\lambda _{k}\mathbf {r} _{k}}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{k}}</annotation>
</semantics>
</math></span><img src="./57dfa9eb1c96d16ff53f264bd9710a16c0108469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.509ex;" alt="{\displaystyle \lambda _{k}}" loading="lazy"></span> is the corresponding <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a>;
the solution of this equation is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t)=\mathbf {r} _{k}e^{\lambda _{k}t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mi>t</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t)=\mathbf {r} _{k}e^{\lambda _{k}t}}</annotation>
</semantics>
</math></span><img src="./99277a07e0840b54687645ed090b7e7920482ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.076ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} (t)=\mathbf {r} _{k}e^{\lambda _{k}t}}" loading="lazy"></span></dd></dl>
<p>as may be confirmed by substitution.
</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>
</semantics>
</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> is <a href="Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a>, then any vector in an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N}</annotation>
</semantics>
</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span>-dimensional space can be represented by a linear combination of the right and <a href="Left_eigenvector" class="mw-redirect" title="Left eigenvector">left eigenvectors</a> (denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {l} _{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">l</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {l} _{k}}</annotation>
</semantics>
</math></span><img src="./43825f50c395bd420a9594760b51ed38473a0127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.831ex; height:2.509ex;" alt="{\displaystyle \mathbf {l} _{k}}" loading="lazy"></span>) of the matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation>
</semantics>
</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span>.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{0}=\sum _{k=1}^{N}\left(\mathbf {l} _{k}\cdot \mathbf {x} _{0}\right)\mathbf {r} _{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">l</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{0}=\sum _{k=1}^{N}\left(\mathbf {l} _{k}\cdot \mathbf {x} _{0}\right)\mathbf {r} _{k}}</annotation>
</semantics>
</math></span><img src="./663d898ed49e0a1b5aac95e5cb2f8aa09dd0d536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.668ex; height:7.343ex;" alt="{\displaystyle \mathbf {x} _{0}=\sum _{k=1}^{N}\left(\mathbf {l} _{k}\cdot \mathbf {x} _{0}\right)\mathbf {r} _{k}}" loading="lazy"></span></dd></dl>
<p>Therefore, the general solution for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t)}</annotation>
</semantics>
</math></span><img src="./385450fed781e07652f160faa271a7cc17fdcbc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.06ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} (t)}" loading="lazy"></span> is
a linear combination of the individual solutions for the right
eigenvectors
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} (t)=\sum _{k=1}^{n}\left(\mathbf {l} _{k}\cdot \mathbf {x} _{0}\right)\mathbf {r} _{k}e^{\lambda _{k}t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">l</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mi>t</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} (t)=\sum _{k=1}^{n}\left(\mathbf {l} _{k}\cdot \mathbf {x} _{0}\right)\mathbf {r} _{k}e^{\lambda _{k}t}}</annotation>
</semantics>
</math></span><img src="./5e016ab16ab0c37b1b0890718790073f81439636.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.99ex; height:6.843ex;" alt="{\displaystyle \mathbf {x} (t)=\sum _{k=1}^{n}\left(\mathbf {l} _{k}\cdot \mathbf {x} _{0}\right)\mathbf {r} _{k}e^{\lambda _{k}t}}" loading="lazy"></span></dd></dl>
<p>Similar considerations apply to the discrete mappings.
</p>
<div class="mw-heading mw-heading2"><h2 id="Classification_in_two_dimensions">Classification in two dimensions</h2></div>
<p>The roots of the <a href="Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> det(<b>A</b> - λ<b>I</b>) are the eigenvalues of <b>A</b>. The sign and relation of these roots, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{n}}</annotation>
</semantics>
</math></span><img src="./093ee22c3daf31b92ff5fa04ba0ce7862283e90c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle \lambda _{n}}" loading="lazy"></span>, to each other may be used to determine the stability of the dynamical system
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {x} (t).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {x} (t).}</annotation>
</semantics>
</math></span><img src="./208049792d1bfa30547a2ae1481b586f89caf4b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.776ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}\mathbf {x} (t)=\mathbf {A} \mathbf {x} (t).}" loading="lazy"></span></dd></dl>
<p>For a 2-dimensional system, the characteristic polynomial is of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{2}-\tau \lambda +\Delta =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>τ<!-- τ --></mi>
<mi>λ<!-- λ --></mi>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda ^{2}-\tau \lambda +\Delta =0}</annotation>
</semantics>
</math></span><img src="./6b76a9680947f3d45524077924415a830b16fc6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.844ex; height:2.843ex;" alt="{\displaystyle \lambda ^{2}-\tau \lambda +\Delta =0}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>
</semantics>
</math></span><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span> is the <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>
</semantics>
</math></span><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"></span> is the <a href="Determinant" title="Determinant">determinant</a> of <b>A</b>. Thus the two roots are in the form:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}={\frac {\tau +{\sqrt {\tau ^{2}-4\Delta }}}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>τ<!-- τ --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>4</mn>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</msqrt>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}={\frac {\tau +{\sqrt {\tau ^{2}-4\Delta }}}{2}}}</annotation>
</semantics>
</math></span><img src="./66817582415854852a191515ab54ead4b5fd5860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.961ex; height:6.176ex;" alt="{\displaystyle \lambda _{1}={\frac {\tau +{\sqrt {\tau ^{2}-4\Delta }}}{2}}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{2}={\frac {\tau -{\sqrt {\tau ^{2}-4\Delta }}}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>τ<!-- τ --></mi>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>4</mn>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
</msqrt>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{2}={\frac {\tau -{\sqrt {\tau ^{2}-4\Delta }}}{2}}}</annotation>
</semantics>
</math></span><img src="./ed487e78a31c439334f17c52c5c3df787287160e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.961ex; height:6.176ex;" alt="{\displaystyle \lambda _{2}={\frac {\tau -{\sqrt {\tau ^{2}-4\Delta }}}{2}}}" loading="lazy"></span>,</dd></dl>
<p>and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\lambda _{1}\lambda _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo>=</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta =\lambda _{1}\lambda _{2}}</annotation>
</semantics>
</math></span><img src="./bd0abe20fdc35e9930365f0d5edc020bfe6f777f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.853ex; height:2.509ex;" alt="{\displaystyle \Delta =\lambda _{1}\lambda _{2}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =\lambda _{1}+\lambda _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
<mo>=</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau =\lambda _{1}+\lambda _{2}}</annotation>
</semantics>
</math></span><img src="./6308b24cc455cd9688c87ee2cc463090c09faad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.96ex; height:2.509ex;" alt="{\displaystyle \tau =\lambda _{1}+\lambda _{2}}" loading="lazy"></span>. Thus if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta <0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo><</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta <0}</annotation>
</semantics>
</math></span><img src="./fcc1b4ee97fd845583ecac5c0a2d151dfac284a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta <0}" loading="lazy"></span> then the eigenvalues are of opposite sign, and the fixed point is a saddle. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta >0}</annotation>
</semantics>
</math></span><img src="./e5bba4bbddf69fb5d000a3d8a9daba0a36b5e720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta >0}" loading="lazy"></span> then the eigenvalues are of the same sign. Therefore, if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
<mo>></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau >0}</annotation>
</semantics>
</math></span><img src="./18720915c5f7ddaefe394e4a0e1012aac186e77f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle \tau >0}" loading="lazy"></span> both are positive and the point is unstable, and if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau <0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
<mo><</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau <0}</annotation>
</semantics>
</math></span><img src="./21d21a211f4467b577c056c1d7df6de8039c3213.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle \tau <0}" loading="lazy"></span> then both are negative and the point is stable. The <a href="Discriminant" title="Discriminant">discriminant</a> will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).
</p><p><br>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Linear_system" title="Linear system">Linear system</a></li>
<li><a href="Dynamical_system" title="Dynamical system">Dynamical system</a></li>
<li><a href="List_of_dynamical_system_topics" class="mw-redirect" title="List of dynamical system topics">List of dynamical system topics</a></li>
<li><a href="Matrix_differential_equation" title="Matrix differential equation">Matrix differential equation</a></li></ul>
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