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INVERSE MATRIX 465
B.4 EIGENVALUES AND EIGENVECTORS OF A MATRIX 2

The eigenvalue or characteristic value and its corresponding eigenvector or char-
acteristic vector of an N × N matrix A aredefined to beascalar λ and a nonzero
vector v satisfying

Av = λv ⇔ (A − λI)v = 0 (v = 0) (B.4.1)

where (λ, v) is called an eigenpair and there are N eigenpairs for an N × N
matrix A.
The eigenvalues of a matrix can be computed as the roots of the characteristic
equation
|A − λI|= 0 (B.4.2)

and the eigenvector corresponding to an eigenvalue λ i can be obtained by sub-
stituting λ i into Eq. (B.4.1) and solve it for v.
Note the following properties.

ž If A is symmetric, all the eigenvalues are real-valued.
ž If A is symmetric and positive definite, all the eigenvalues are real and
positive.
ž If v is an eigenvector of A,sois cv for any nonzero scalar c.

B.5 INVERSE MATRIX
The inverse matrix of a K × K (square) matrix A = [a mn ] is denoted by A −1
and defined to be a matrix which is premultiplied/postmultiplied by A to form
an identity matrix—that is, satisfies
A × A −1 = A −1 × A = I (B.5.1)

An element of the inverse matrix A −1 = [α mn ] can be computed as
1 1 m+n
α mn = A mn = (−1) M mn (B.5.2)
det(A) |A|

where M kn is the minor of a kn and A kn = (−1) k+n M kn is the cofactor
of a kn .

2 See the website @http://www.sosmath.com/index.html or http://www.psc.edu/∼burkardt/papers/
linear glossary.html.)