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Modal matrix
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<h2 id="example">Example</h2> <p>The matrix </p> A = ( 3 2 0 2 0 0 1 0 2 ) {\displaystyle A={\begin{pmatrix}3&2&0\\2&0&0\\1&0&2\end{pmatrix}}} <p>has eigenvalues and corresponding eigenvectors </p> λ 1 = − 1 , b 1 = ( − 3 , 6 , 1 ) , {\displaystyle \lambda _{1}=-1,\quad \,\mathbf {b} _{1}=\left(-3,6,1\right),} λ 2 = 2 , b 2 = ( 0 , 0 , 1 ) , {\displaystyle \lambda _{2}=2,\qquad \mathbf {b} _{2}=\left(0,0,1\right),} λ 3 = 4 , b 3 = ( 2 , 1 , 1 ) . {\displaystyle \lambda _{3}=4,\qquad \mathbf {b} _{3}=\left(2,1,1\right).} <p>A diagonal matrix D {\displaystyle D} , <a href="/facts/Matrix_similarity/ySevpav1">similar</a> to A {\displaystyle A} is </p> D = ( − 1 0 0 0 2 0 0 0 4 ) . {\displaystyle D={\begin{pmatrix}-1&0&0\\0&2&0\\0&0&4\end{pmatrix}}.} <p>One possible choice for an <a href="/facts/Invertible_matrix/fPqXk3V8">invertible matrix</a> M {\displaystyle M} such that D = M − 1 A M , {\displaystyle D=M^{-1}AM,} is </p> M = ( − 3 0 2 6 0 1 1 1 1 ) . {\displaystyle M={\begin{pmatrix}-3&0&2\\6&0&1\\1&1&1\end{pmatrix}}.} <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> <p>Note that since eigenvectors themselves are not unique, and since the columns of both M {\displaystyle M} and D {\displaystyle D} may be interchanged, it follows that both M {\displaystyle M} and D {\displaystyle D} are not unique.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="generalized-modal-matrix">Generalized modal matrix</h2> <p>Let A {\displaystyle A} be an <i>n</i> × <i>n</i> matrix. A generalized modal matrix M {\displaystyle M} for A {\displaystyle A} is an <i>n</i> × <i>n</i> matrix whose columns, considered as vectors, form a <a href="/facts/Canonical_basis/I3dJq0Q3">canonical basis</a> for A {\displaystyle A} and appear in M {\displaystyle M} according to the following rules: </p> <ul><li>All <a href="/facts/Jordan_chain/Lf7MgEGJ">Jordan chains</a> consisting of one vector (that is, one vector in length) appear in the first columns of M {\displaystyle M} .</li> <li>All vectors of one chain appear together in adjacent columns of M {\displaystyle M} .</li> <li>Each chain appears in M {\displaystyle M} in order of increasing rank (that is, the <a href="/facts/Generalized_eigenvector/Lf7MgEGJ">generalized eigenvector</a> of rank 1 appears before the generalized eigenvector of rank 2 of the same chain, which appears before the generalized eigenvector of rank 3 of the same chain, etc.).<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul> <p>One can show that </p> <table><tbody><tr><td> A M = M J , {\displaystyle AM=MJ,} </td> <td></td> <td>1</td></tr></tbody></table> <p>where J {\displaystyle J} is a matrix in <a href="/facts/Jordan_normal_form/M7ezYbnl">Jordan normal form</a>. By premultiplying by M − 1 {\displaystyle M^{-1}} , we obtain </p> <table><tbody><tr><td> J = M − 1 A M . {\displaystyle J=M^{-1}AM.} </td> <td></td> <td>2</td></tr></tbody></table> <p>Note that when computing these matrices, equation (1) is the easiest of the two equations to verify, since it does not require <a href="/facts/Invertible_matrix/fPqXk3V8">inverting</a> a matrix.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h3>Example</h3> <p>This example illustrates a generalized modal matrix with four Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> The matrix </p> A = ( − 1 0 − 1 1 1 3 0 0 1 0 0 0 0 0 2 1 2 − 1 − 1 − 6 0 − 2 0 − 1 2 1 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 − 1 − 1 0 1 2 4 1 ) {\displaystyle A={\begin{pmatrix}-1&0&-1&1&1&3&0\\0&1&0&0&0&0&0\\2&1&2&-1&-1&-6&0\\-2&0&-1&2&1&3&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\-1&-1&0&1&2&4&1\end{pmatrix}}} <p>has a single eigenvalue λ 1 = 1 {\displaystyle \lambda _{1}=1} with <a href="/facts/Algebraic_multiplicity/8TjEoT8u">algebraic multiplicity</a> μ 1 = 7 {\displaystyle \mu _{1}=7} . A canonical basis for A {\displaystyle A} will consist of one linearly independent generalized eigenvector of rank 3 (generalized eigenvector rank; see <a href="/facts/Generalized_eigenvector/Lf7MgEGJ">generalized eigenvector</a>), two of rank 2 and four of rank 1; or equivalently, one chain of three vectors { x 3 , x 2 , x 1 } {\displaystyle \left\{\mathbf {x} _{3},\mathbf {x} _{2},\mathbf {x} _{1}\right\}} , one chain of two vectors { y 2 , y 1 } {\displaystyle \left\{\mathbf {y} _{2},\mathbf {y} _{1}\right\}} , and two chains of one vector { z 1 } {\displaystyle \left\{\mathbf {z} _{1}\right\}} , { w 1 } {\displaystyle \left\{\mathbf {w} _{1}\right\}} . </p><p>An "almost diagonal" matrix J {\displaystyle J} in <i>Jordan normal form</i>, similar to A {\displaystyle A} is obtained as follows: </p> M = ( z 1 w 1 x 1 x 2 x 3 y 1 y 2 ) = ( 0 1 − 1 0 0 − 2 1 0 3 0 0 1 0 0 − 1 1 1 1 0 2 0 − 2 0 − 1 0 0 − 2 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 − 1 0 − 1 0 ) , {\displaystyle M={\begin{pmatrix}\mathbf {z} _{1}&\mathbf {w} _{1}&\mathbf {x} _{1}&\mathbf {x} _{2}&\mathbf {x} _{3}&\mathbf {y} _{1}&\mathbf {y} _{2}\end{pmatrix}}={\begin{pmatrix}0&1&-1&0&0&-2&1\\0&3&0&0&1&0&0\\-1&1&1&1&0&2&0\\-2&0&-1&0&0&-2&0\\1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&0&-1&0&-1&0\end{pmatrix}},} J = ( 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 ) , {\displaystyle J={\begin{pmatrix}1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&1&1&0&0&0\\0&0&0&1&1&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&1\\0&0&0&0&0&0&1\end{pmatrix}},} <p>where M {\displaystyle M} is a generalized modal matrix for A {\displaystyle A} , the columns of M {\displaystyle M} are a canonical basis for A {\displaystyle A} , and A M = M J {\displaystyle AM=MJ} .<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both M {\displaystyle M} and J {\displaystyle J} may be interchanged, it follows that both M {\displaystyle M} and J {\displaystyle J} are not unique.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h2 id="notes">Notes</h2> <ul><li>Beauregard, Raymond A.; Fraleigh, John B. (1973), <a href="https://archive.org/details/firstcourseinlin0000beau"><i>A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields</i></a>, Boston: <a href="/facts/Houghton_Mifflin_Co./byG66zxw">Houghton Mifflin Co.</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-395-14017-X</li> <li>Bronson, Richard (1970), <i>Matrix Methods: An Introduction</i>, New York: <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/70097490">70097490</a></li> <li>Nering, Evar D. (1970), <i>Linear Algebra and Matrix Theory</i> (2nd ed.), New York: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">Wiley</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/76091646">76091646</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Bronson (1970, pp. 179–183) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bronson (1970, p. 181) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Beauregard & Fraleigh (1973, pp. 271, 272) - Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X <a href="https://archive.org/details/firstcourseinlin0000beau" target="_blank">https://archive.org/details/firstcourseinlin0000beau</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Bronson (1970, p. 181) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Bronson (1970, p. 205) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Bronson (1970, pp. 206–207) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Nering (1970, pp. 122, 123) - Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 <a href="https://lccn.loc.gov/76091646" target="_blank">https://lccn.loc.gov/76091646</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Bronson (1970, pp. 208, 209) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Bronson (1970, p. 206) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 <a href="https://lccn.loc.gov/70097490" target="_blank">https://lccn.loc.gov/70097490</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>