Modal matrix

In linear algebra, the modal matrix is used in the diagonalization process involving eigenvalues and eigenvectors.^[1]

Specifically the modal matrix M {\displaystyle M} for the matrix A {\displaystyle A} is the n × n matrix formed with the eigenvectors of A {\displaystyle A} as columns in M {\displaystyle M} . It is utilized in the similarity transformation

D = M − 1 A M , {\displaystyle D=M^{-1}AM,}

where D {\displaystyle D} is an n × n diagonal matrix with the eigenvalues of A {\displaystyle A} on the main diagonal of D {\displaystyle D} and zeros elsewhere. The matrix D {\displaystyle D} is called the spectral matrix for A {\displaystyle A} . The eigenvalues must appear left to right, top to bottom in the same order as their corresponding eigenvectors are arranged left to right in M {\displaystyle M} .^[2]

The matrix

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}}}

has eigenvalues and corresponding eigenvectors

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).}

A diagonal matrix D {\displaystyle D} , similar to A {\displaystyle A} is

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}}.}

One possible choice for an invertible matrix M {\displaystyle M} such that D = M − 1 A M , {\displaystyle D=M^{-1}AM,} is

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}}.} ^[3]

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.^[4]

Let A {\displaystyle A} be an n × n matrix. A generalized modal matrix M {\displaystyle M} for A {\displaystyle A} is an n × n matrix whose columns, considered as vectors, form a canonical basis for A {\displaystyle A} and appear in M {\displaystyle M} according to the following rules:

• All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of M {\displaystyle M} .
• All vectors of one chain appear together in adjacent columns of M {\displaystyle M} .
• Each chain appears in M {\displaystyle M} in order of increasing rank (that is, the generalized eigenvector 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.).^[5]

One can show that

A M = M J , {\displaystyle AM=MJ,}

1

where J {\displaystyle J} is a matrix in Jordan normal form. By premultiplying by M − 1 {\displaystyle M^{-1}} , we obtain

J = M − 1 A M . {\displaystyle J=M^{-1}AM.}

2

Note that when computing these matrices, equation (1) is the easiest of the two equations to verify, since it does not require inverting a matrix.^[6]

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.^[7] The matrix

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}}}

has a single eigenvalue λ 1 = 1 {\displaystyle \lambda _{1}=1} with algebraic multiplicity 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 generalized eigenvector), 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\}} .

An "almost diagonal" matrix J {\displaystyle J} in Jordan normal form, similar to A {\displaystyle A} is obtained as follows:

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}},}

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} .^[8] 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.^[9]

• 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
• Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646