1.1 Basic Concepts and Formulae 15
Equivalence
AandBare 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
IfA=[aij] is a square matrix andαijthe cofactor ofaijthen
adjA=⎡
⎢
⎢
⎣
α 11 α 21 ··· αn 1
α 12 α 22 ··· ···
··· ··· ··· ···
··· ··· ··· αnn⎤
⎥
⎥
⎦
The cofactorαij=(−1)i+jMij
whereMijis the minor obtained by striking off theith row andjth 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
LetAX=λX (1.84)be the transformation of the vectorXintoλX, whereλis a number, thenλis called
the eigen or characteristic value.
From (1.84):
(A−λI)X=⎡
⎢
⎢
⎢
⎣
a 11 −λ a 12 ··· a 1 n
a 21 a 22 −λ··· a 2 n
... ··· ··· ···
an 1 ··· ···ann−λ
⎤
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎣
x 1
x 2
..
.
xn