1000 Solved Problems in Modern Physics

(Tina Meador) #1

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






= 0 (1.85)
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