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modal matrix ：ウィキペディア英語版

modal matrix
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 transformation
:

D = M^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 3 & 2 & 0 \\2 & 0 & 0 \\1 & 0 & 2\end

has eigenvalues and corresponding eigenvectors
:

\lambda_1 = -1, \quad \, \bold b_1 = \left( -3, 6, 1 \right) ,

\lambda_2 = 2, \qquad \bold b_2 = \left( 0, 0, 1 \right) ,

\lambda_3 = 4, \qquad \bold b_3 = \left( 2, 1, 1 \right) .

A diagonal matrix

D

, similar to

A

is
:

D = \begin -1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 4\end.

One possible choice for an invertible matrix

M

such that

D = M^AM,

is
:

M = \begin -3 & 0 & 2 \\6 & 0 & 1 \\1 & 1 & 1\end.

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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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