6.5 Matrix Factorization (Cont’d)
Example. Consider matrix
[
] Matrix
[
Permutation Matrices
Definition. Suppose
[
], which is obtained by interchanging the 2nd and 3rd rows of identity matrix
]. What is
?
is a permutation of
. The permutation matrix
is defined by
{

permutes the rows of
[
]

exists and
Gaussian elimination with row interchanges can be written as:
Remark:
is not lower triangular matrix unless
Example. Find a factorization
is identity matrix.
for the matrix
[
]
1
6.6 Special Types of Matrices
Definition. The
is said to be strictly diagonally dominant when |
matrix
Example. Determine if matrices
[
], and
[
|
∑
|
| holds for each
] are strictly diagonally dominant.
Theorem 6.21. A strictly diagonally dominant matrix is nonsingular, Gaussian elimination can be performed on
without row interchanges. The computations will be stable with respect to the growth of round-off errors.
Definition. Matrix
is said to be positive definite if it is symmetric and if
Example. Show that the matrix
Solution: Let
[
for every nonzero vector
(i.e.
)
] is positive definite.
be a 3-dimentional column vector.
[
][ ]
unless
Theorem 6.23. If is an
positive definite matrix, then
(i)
has an inverse
(ii)
for each
(iii)
| |
| |
(iv)
for each
2
Definition. A leading principal submatrix of a matrix
for some
is a matrix of the form
[
]
.
Theorem 6.25. A symmetric matrix
determinant.
is positive definite if and only if each of its leading principal submatrix has a positive
Example. Show all leading principle submatrix of
[
] have positive determinants.
3