Page 232 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
C 0
1 C
Thus every 2 × 2 matrix over the field of complex numbers is similar to a matrix of one of the two
types displayed above, possibly with C = C .
1 2

Example 1: Let T be represented in ordered basis by the matrix

0 1 1
A 0 0 0 F
3
0 0 0
The ordered basis is  = (1, 0, 0),  = (0, 1, 0),  = (0, 0, 1)
1 2 3
Let  =  ,  = A =  +  ,  =  . In this basis
1 1 2 1 2 3 3 3
( ,  ,  ) the matrix A becomes
1 2 3
A’ = PAP –1

1 0 0
where P = 0 1 1 ,
0 0 1

A straight forward method gives

1 0 0
P = 0 1 1 ,
–1
0 0 1
0 1 0
then A’ = 0 0 0
0 0 0

which is in Jordan form. Thus A is similar to A’.

Example 2: Let A be a complex 3 × 3 matrix

2 0 0
A a 2 0
b c 1

2
The characteristic polynomial for A is obviously (x – 2) (x + 1). Either this is the minimal
polynomial, in which case A is similar to

2 0 0
1 2 0
0 0 1

or the minimal polynomial is (x – 2) (x + 1), in which case A is similar to

2 0 0
0 2 0
0 0 1

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