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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Matrix polynomial</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Polynomial_matrix" title="Polynomial matrix">Polynomial matrix</a>.</div>
<p>In mathematics, a <b>matrix polynomial</b> is a polynomial with <a href="Square_matrix" title="Square matrix">square matrices</a> as variables. Given an ordinary, scalar-valued polynomial
</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 P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}">
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<annotation encoding="application/x-tex">{\displaystyle P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}</annotation>
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</math></span><img src="./fedf00e6d1e4f4c50a92e35bf3fac3c489d1eef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.276ex; height:6.843ex;" alt="{\displaystyle P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}" loading="lazy"></span></dd></dl>
<p>this polynomial evaluated at a 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 A}">
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<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> 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 P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>P</mi>
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<annotation encoding="application/x-tex">{\displaystyle P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},}</annotation>
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</math></span><img src="./66dc9788b5ccf460b7cb64a24fad0592e552ce66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.515ex; height:6.843ex;" alt="{\displaystyle P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},}" 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 I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>
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</math></span><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> is the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>.<sup id="cite_ref-FOOTNOTEHornJohnson199036_1-0" class="reference"><a href="#cite_note-FOOTNOTEHornJohnson199036-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Note that <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 P(A)}">
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<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation>
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</math></span><img src="./4f264d19e21604793c6dc54f8044df454db82744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}" loading="lazy"></span> has the same dimension as <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 A}">
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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>.
</p><p>A <b>matrix polynomial equation</b> is an equality between two matrix polynomials, which holds for the specific matrices in question. A <b>matrix polynomial identity</b> is a matrix polynomial equation which holds for all matrices <i>A</i> in a specified <a href="Matrix_ring" title="Matrix ring">matrix ring</a> <i>M<sub>n</sub></i>(<i>R</i>).
</p><p>Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of <a href="Linear_transformations" class="mw-redirect" title="Linear transformations">linear transformations</a> represented as matrices, most notably the <a href="Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a>.
</p>
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<div class="mw-heading mw-heading2"><h2 id="Characteristic_and_minimal_polynomial">Characteristic and minimal polynomial</h2></div>
<p>The <a href="Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of a matrix <i>A</i> is a scalar-valued polynomial, defined 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 p_{A}(t)=\det \left(tI-A\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
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<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle p_{A}(t)=\det \left(tI-A\right)}</annotation>
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</math></span><img src="./616d5a94f2fa3fc44245832fc1ffc36517eb9f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:20.492ex; height:2.843ex;" alt="{\displaystyle p_{A}(t)=\det \left(tI-A\right)}" loading="lazy"></span>. The <a href="Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a> states that if this polynomial is viewed as a matrix polynomial and evaluated at 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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> itself, the result is the zero 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 p_{A}(A)=0}">
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<annotation encoding="application/x-tex">{\displaystyle p_{A}(A)=0}</annotation>
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</math></span><img src="./72a0eb1479ce148c3d8e2722c35418dcb1289057.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.537ex; height:2.843ex;" alt="{\displaystyle p_{A}(A)=0}" loading="lazy"></span>. A polynomial <i>annihilates</i> <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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> 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 p(A)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle p(A)=0}</annotation>
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</math></span><img src="./f85400cd1f468ae64bfab866dac4618474c55356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.072ex; height:2.843ex;" alt="{\displaystyle p(A)=0}" loading="lazy"></span>; <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 p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> is also known as an <i><a href="Annihilating_polynomial" title="Annihilating polynomial">annihilating polynomial</a></i>. Thus, the characteristic polynomial is a polynomial which annihilates <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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>.
</p><p>There is a unique <a href="Monic_polynomial" title="Monic polynomial">monic polynomial</a> of minimal degree which annihilates <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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>; this polynomial is the <a href="Minimal_polynomial_(linear_algebra)" title="Minimal polynomial (linear algebra)">minimal polynomial</a>. Any polynomial which annihilates <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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> (such as the characteristic polynomial) is a multiple of the minimal polynomial.<sup id="cite_ref-FOOTNOTEHornJohnson1990Thm_3.3.1_2-0" class="reference"><a href="#cite_note-FOOTNOTEHornJohnson1990Thm_3.3.1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>It follows that given two polynomials <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 P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" 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 Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span>, we have <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 P(A)=Q(A)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Q</mi>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(A)=Q(A)}</annotation>
</semantics>
</math></span><img src="./fb72f0917f05561dca07fc499a0820dd21a26ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.787ex; height:2.843ex;" alt="{\displaystyle P(A)=Q(A)}" loading="lazy"></span> if and only if
</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 P^{(j)}(\lambda _{i})=Q^{(j)}(\lambda _{i})\qquad {\text{for }}j=0,\ldots ,n_{i}-1{\text{ and }}i=1,\ldots ,s,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mspace width="2em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mtext>for </mtext>
</mrow>
<mi>j</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mtext> and </mtext>
</mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>s</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P^{(j)}(\lambda _{i})=Q^{(j)}(\lambda _{i})\qquad {\text{for }}j=0,\ldots ,n_{i}-1{\text{ and }}i=1,\ldots ,s,}</annotation>
</semantics>
</math></span><img src="./02f455cd44acbd10da2c02dbf507056a14d2018d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.559ex; height:3.343ex;" alt="{\displaystyle P^{(j)}(\lambda _{i})=Q^{(j)}(\lambda _{i})\qquad {\text{for }}j=0,\ldots ,n_{i}-1{\text{ and }}i=1,\ldots ,s,}" 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 P^{(j)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P^{(j)}}</annotation>
</semantics>
</math></span><img src="./f980d565cf562e20eccc89072d66d20b6a7a78a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.011ex; height:2.843ex;" alt="{\displaystyle P^{(j)}}" loading="lazy"></span> denotes the <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 j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
</semantics>
</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>th derivative of <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 P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" 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 \lambda _{1},\dots ,\lambda _{s}}">
<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>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\dots ,\lambda _{s}}</annotation>
</semantics>
</math></span><img src="./dd26e1c969db141dcb38c3b817be0c960a0c1efb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.946ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\dots ,\lambda _{s}}" loading="lazy"></span> are the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of <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 A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> with corresponding indices <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_{1},\dots ,n_{s}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>s</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n_{1},\dots ,n_{s}}</annotation>
</semantics>
</math></span><img src="./756a77dfc09fa9f3910f421ddec0feea2f0c315c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.025ex; height:2.009ex;" alt="{\displaystyle n_{1},\dots ,n_{s}}" loading="lazy"></span> (the index of an eigenvalue is the size of its largest <a href="Jordan_normal_form" title="Jordan normal form">Jordan block</a>).<sup id="cite_ref-FOOTNOTEHigham2000Thm_1.3_3-0" class="reference"><a href="#cite_note-FOOTNOTEHigham2000Thm_1.3-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Matrix_geometrical_series">Matrix geometrical series</h2></div>
<p>Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary <a href="Geometric_series" title="Geometric series">geometric series</a>,
</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 S=I+A+A^{2}+\cdots +A^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mi>I</mi>
<mo>+</mo>
<mi>A</mi>
<mo>+</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S=I+A+A^{2}+\cdots +A^{n}}</annotation>
</semantics>
</math></span><img src="./a6846ce01770bedacc3a162446849032bbc54e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.356ex; height:2.843ex;" alt="{\displaystyle S=I+A+A^{2}+\cdots +A^{n}}" 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 AS=A+A^{2}+A^{3}+\cdots +A^{n+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mi>S</mi>
<mo>=</mo>
<mi>A</mi>
<mo>+</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle AS=A+A^{2}+A^{3}+\cdots +A^{n+1}}</annotation>
</semantics>
</math></span><img src="./7298eba243445186f446a0418f074c272d691fbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:32.826ex; height:2.843ex;" alt="{\displaystyle AS=A+A^{2}+A^{3}+\cdots +A^{n+1}}" 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 (I-A)S=S-AS=I-A^{n+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mi>S</mi>
<mo>=</mo>
<mi>S</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mi>S</mi>
<mo>=</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (I-A)S=S-AS=I-A^{n+1}}</annotation>
</semantics>
</math></span><img src="./29b1018a42a08ba4b40fe17f54ebeec798d60ab7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.917ex; height:3.176ex;" alt="{\displaystyle (I-A)S=S-AS=I-A^{n+1}}" 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 S=(I-A)^{-1}(I-A^{n+1})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<msup>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S=(I-A)^{-1}(I-A^{n+1})}</annotation>
</semantics>
</math></span><img src="./785b1b09ef8cfaa80493ffcd46387a4483276cb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.379ex; height:3.176ex;" alt="{\displaystyle S=(I-A)^{-1}(I-A^{n+1})}" loading="lazy"></span></dd></dl>
<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 I-A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I-A}</annotation>
</semantics>
</math></span><img src="./9274b91f0029162df383d65384d7748a2ef3d4cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.755ex; height:2.343ex;" alt="{\displaystyle I-A}" loading="lazy"></span> is nonsingular one can evaluate the expression for the sum <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 S}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S}</annotation>
</semantics>
</math></span><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Latimer%E2%80%93MacDuffee_theorem" title="Latimer–MacDuffee theorem">Latimer–MacDuffee theorem</a></li>
<li><a href="Matrix_exponential" title="Matrix exponential">Matrix exponential</a></li>
<li><a href="Matrix_function" class="mw-redirect" title="Matrix function">Matrix function</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
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.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}
/* end https://en.wikipedia.org/ */
</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-FOOTNOTEHornJohnson199036-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHornJohnson199036_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHornJohnson1990">Horn & Johnson 1990</a>, p. 36.</span>
</li>
<li id="cite_note-FOOTNOTEHornJohnson1990Thm_3.3.1-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHornJohnson1990Thm_3.3.1_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHornJohnson1990">Horn & Johnson 1990</a>, Thm 3.3.1.</span>
</li>
<li id="cite_note-FOOTNOTEHigham2000Thm_1.3-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHigham2000Thm_1.3_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHigham2000">Higham 2000</a>, Thm 1.3.</span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFGohbergLancasterRodman2009" class="citation book cs1">Gohberg, Israel; <a href="Peter_Lancaster" title="Peter Lancaster">Lancaster, Peter</a>; Rodman, Leiba (2009) [1982]. <i>Matrix Polynomials</i>. Classics in Applied Mathematics. Vol. 58. Lancaster, PA: <a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-898716-81-8</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1170.15300">1170.15300</a>.</cite></li>
<li><cite id="CITEREFHigham2000" class="citation book cs1">Higham, Nicholas J. (2000). <i>Functions of Matrices: Theory and Computation</i>. SIAM. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>089-871-777-9</bdi>.</cite>.</li>
<li><cite id="CITEREFHornJohnson1990" class="citation book cs1">Horn, Roger A.; Johnson, Charles R. (1990). <i>Matrix Analysis</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-38632-6</bdi>.</cite>.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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