Biomedical Engineering Reference
In-Depth Information
Regarding the matrix determinant, we have the following identity,
=|
AB
CD
CA
−
1
B
BD
−
1
C
A
||
D
−
|=|
D
||
A
−
|
.
(C.94)
When a matrix
A
is an invertible
(
n
×
n
)
matrix, and
B
and
C
are
(
n
×
m
)
matrices,
the following matrix determinant lemma holds:
B
T
A
−
1
C
CB
T
|
A
||
I
+
|=|
A
+
|
.
(C.95)
C.8
Properties of Eigenvalues
Representative properties of eigenvalues that may be used in this topic are listed
below.
1. If
A
is a Hermitian matrix and positive definite, all eigenvalues are real and
greater than 0.
2. If
A
is a Hermitian matrix and positive semidefinite, all eigenvalues are real and
greater than or equal to 0.
3. If
A
is a real symmetric matrix and positive definite, all eigenvalues are real and
greater than 0. Eigenvectors are also real.
4. If
A
is a real symmetric matrix and positive semidefinite, all eigenvalues are real
and greater than or equal to 0. Eigenvectors are also real.
5. If
A
and
B
are square matrices and
B
is nonsingular, eigenvalues of
A
are also
eigenvalues of
B
−
1
AB
.
6. If
A
and
B
are square matrices and
B
is unitary, eigenvalues of
A
are also
eigenvalues of
B
H
AB
.
7. If
A
and
B
are square matrices and
B
is orthogonal, eigenvalues of
A
are also
eigenvalues of
B
T
AB
.
8. If
A
and
B
are square matrices and
B
is positive definite, eigenvalues of
BA
are
also eigenvalues of
B
1
/
2
AB
1
/
2
.
9. If
A
and
B
are square matrices and
B
is positive definite, eigenvalues of
B
−
1
A
are also eigenvalues of
B
−
1
/
2
AB
−
1
/
2
.
10. Let us assume that
A
is an
(
m
×
n
)
matrix,
B
is an
(
n
×
m
)
matrix, and
n
≥
m
.If
ʻ
1
,...,ʻ
m
are eigenvalues of
AB
,
ʻ
1
,...,ʻ
m
,
0
,...,
0 are eigenvalues of
BA
.
C.9
Rayleigh-Ritz Formula
This section provides a proof of the Rayleigh-Ritz formula, which is according to [2].
We define
A
and
B
as positive definite matrices of the same dimension. We introduce
the following notations and use them throughout the topic:
