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
−
1
A
M
,
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
.
The matrix
A
=
(
3
2
0
2
0
0
1
0
2
)
has eigenvalues and corresponding eigenvectors
λ
1
=
−
1
,
b
1
=
(
−
3
,
6
,
1
)
,
λ
2
=
2
,
b
2
=
(
0
,
0
,
1
)
,
λ
3
=
4
,
b
3
=
(
2
,
1
,
1
)
.
A diagonal matrix
D
, similar to
A
is
D
=
(
−
1
0
0
0
2
0
0
0
4
)
.
One possible choice for an invertible matrix
M
such that
D
=
M
−
1
A
M
,
is
M
=
(
−
3
0
2
6
0
1
1
1
1
)
.
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.
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 (1) is the easiest of the two equations to verify, since it does not require inverting a matrix.
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
=
(
−
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
)
has a single eigenvalue
λ
1
=
1
with algebraic multiplicity
μ
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
{
x
3
,
x
2
,
x
1
}
, one chain of two vectors
{
y
2
,
y
1
}
, and two chains of one vector
{
z
1
}
,
{
w
1
}
.
An "almost diagonal" matrix
J
in Jordan normal form, similar to
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
)
,
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
)
,
where
M
is a generalized modal matrix for
A
, the columns of
M
are a canonical basis for
A
, and
A
M
=
M
J
. 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.
