1.1 Basic Concepts and Formulae
15
Equivalence
A and B are said to be equivalent ( A ∼ B) if one can be obtained from the other by
a sequence of elementary transformations.
The adjoint of a square matrix
If A = [a i j ] is a square matrix and α i j the cofactor of a i j then
adj A =
⎡
⎢
⎢
⎣
α 11 α 21 · · · α n1
α 12 α 22 · · · · · ·
· · · · · · · · · · · ·
· · · · · · · · · α nn
⎤
⎥
⎥
⎦
The cofactor α i j = (−1)
i+ j M i j
where M i j is the minor obtained by striking off the ith row and jth column and
computing the determinant from the remaining elements.
Inverse from the adjoint
A
−1
=
ad j A
|A|
Inverse for orthogonal matrices
A
−1
= A
Inverse of unitary matrices
A
−1
= ( A)
Characteristic equation
Let AX = λX
(1.84)
be the transformation of the vector X into λX , where λ is a number, then λ is called
the eigen or characteristic value.
From (1.84):
(A − λI )X =
⎡
⎢
⎢
⎢
⎣
a 11 − λ a 12 · · · a 1n
a 21 a 22 − λ · · · a 2n
. . .
· · · · · · · · ·
a n1
· · · · · · a nn − λ
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎣
x 1
x 2
. . .
x n
⎤
⎥
⎥
⎥
⎦
= 0
(1.85)