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5.2. MATRICES AND DETERMINANTS 181

These elementary transformations are accomplished with the help of elementary matri-
ces obtained from the unit matrix by the corresponding operations with its rows (columns).
With the help of elementary transformations, an arbitrary matrix A of rank r > 0 can be
reduced to normal (canonical) form, which has a block structure with the unit matrix I of
size r × r in the top left corner.
Example 1. The LU-decomposition of a matrix
( 214 )
A = 321
133

can be obtained with the help of the following sequence of elementary transformations:
S 1 S 2 S 3

( 1/200 )( 214 ) ( 1 0 0 )( 11/2 2 ) ( 1 0 0 )( 11/2 2 )
0 1 0 321 → –310 3 2 1 → 0 1 0 01/2 –5 →
0 0 1 133 0 0 1 1 3 3 –1 0 1 1 3 3
T 1 T 2

11/2 2 1 –1/2 0 1 0 2 10 –2
( ) ( ) ( ) ( )
01/2 –5 0 1 0 1/2 –5 01 0
→ 0 → →
05/2 1 0 0 1 0 5/2 1 00 1
S 4 S 5
10 0 1 0 0 1 0 0 1 0 0

( )( ) ( )( )
→ 02 0 01/2 –5 → 0 1 0 0 1 –10 →
00 1 05/2 1 0 –5/2 1 05/2 1
T 3 S 6

10 0 10 0 10 0 10 0 100
( ) ( ) ( )( ) ( )
01 –10 0110 → 01 0 01 0 010
→ → .
00 26 00 1 001/26 0026 001
These transformations amount to the equivalence transformation I = SAT,where T = T 1T 2T 3:
1/2 0 0 1 –1/2 –7
( ) ( )
S = S 6S 5S 4S 3S 2S 1 = –3 2 0 and T = T 1T 2T 3 = 0 1 10 .
7/26 –5/26 1/26 0 0 1
Hence, we obtain
2 0 0 11/2 2
( ) ( )
–1 –1
L = S = 31/2 0 and U = T = 0 1 –10 .
15/2 26 0 0 1
5.2.3-2. Similarity transformation.

Two square matrices A and A of the same size are said to be similar if there exists a square
2
nondegenerate matrix S of the same size, the so-called transforming matrix, such that A
and A are related by the similarity transformation
2
–1 –1
A = S AS or A = SAS .
2
2
Properties of similar matrices:
1. If A and B are square matrices of the same size and C = A + B,then
–1
–1
–1
C = A + B or S (A + B)S = S AS + S BS.
2
2
2