Page 479 - Applied Numerical Methods Using MATLAB
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468 MATRIX OPERATIONS/PROPERTIES
Any matrix, singular or rectangular, can be transformed into this form through
the Gaussian elimination procedure (i.e., a series of elementary row operations)
or, equivalently, by using the MATLAB built-in routine “rref()”. For example,
we have
001 3 240 −8
row
A = 240 −8 −−−→ 001 3
121 −1 change 121 −1
row 120 −4 1 2 0 −4
division row
−−−−−→ 001 3 −−−−−→ 0 0 1 3 = rref(A)
row subtraction
subtraction 001 3 0 0 0 0
Once this form is obtained, it is easy to compute the rank, the determinant and
the inverse of the matrix, if only the matrix is invertible.
B.12 POSITIVE DEFINITENESS
A square matrix A is said to be positive definite if
∗T
x Ax > 0 for any nonzero vector x (B.12.1)
A square matrix A is said to be positive semidefinite if
∗T
x Ax ≥ 0 for any nonzero vector x (B.12.2)
Note the following properties of a positive definite matrix A.
ž A is nonsingular and all of its eigenvalues are positive.
ž The inverse of A is also positive definite.
There are similar definitions for negative definiteness and negative semidefinite-
ness.
Note the following property, which can be used to determine if a matrix
is positive (semi-) definite or not. A square matrix is positive definite if and
only if:
(i) Every diagonal element is positive.
(ii) Every leading principal minor matrix has positive determinant.
On the other hand, a square matrix is positive semidefinite if and only if:
(i) Every diagonal element is nonnegative.
(ii) Every principal minor matrix has nonnegative determinant.
