Modal matrix explained

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

Specifically the modal matrix

M for the matrix A is the n × n matrix formed with the eigenvectors of A as columns in M . It is utilized in the similarity transformationD = M ^{-1} AM,

where

D is an n × n diagonal matrix with the eigenvalues of A on the main diagonal of D and zeros elsewhere. The matrix D is called the spectral matrix for 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 .

Example

The matrix

A = \begin{pmatrix} 3 & 2 & 0 \\ 2 & 0 & 0 \\ 1 & 0 & 2 \end{pmatrix}

has eigenvalues and corresponding eigenvectors

λ _{1} = -1 , b _{1} = \left( -3 , 6 , 1 \right) , λ _{2} = 2 , b _{2} = \left( 0 , 0 , 1 \right) , λ _{3} = 4 , b _{3} = \left( 2 , 1 , 1 \right) .

A diagonal matrix

D , similar to A isD = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix}. M such that D = M ^{-1} AM, isM = \begin{pmatrix} -3 & 0 & 2 \\ 6 & 0 & 1 \\ 1 & 1 & 1 \end{pmatrix}.

Note that since eigenvectors themselves are not unique, and since the columns of both

M and D may be interchanged, it follows that both M and D are not unique.

Generalized modal matrix

Let

A be an n × n matrix. A generalized modal matrix M for A is an n × n matrix whose columns, considered as vectors, form a canonical basis for A and appear in 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 .

• All vectors of one chain appear together in adjacent columns of

M .

• Each chain appears in

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.).

One can show that

where

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

Note that when computing these matrices, equation is the easiest of the two equations to verify, since it does not require inverting a matrix.

Example

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.The matrix

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 with algebraic multiplicity \mu _{1} = 7 . A canonical basis for 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 \left\{ x _{3,} x _{2,} x _{1} \right\} , one chain of two vectors \left\{ y _{2,} y _{1} \right\} , and two chains of one vector \left\{ z _{1} \right\} , \left\{ w _{1} \right\} .

An "almost diagonal" matrix

J in Jordan normal form, similar to A is obtained as follows:M = \begin{pmatrix} z _{1} & w _{1} & x _{1} & x _{2} & x _{3} & y _{1} & 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 = \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 is a generalized modal matrix for A , the columns of M are a canonical basis for A , and AM = MJ . Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both M and J may be interchanged, it follows that both M and J are not unique