Page 137 - DMTH502_LINEAR_ALGEBRA

Unit 9: Representation of Transformations by Matrices

Dg 2g Notes
3 2
Dg 3 .
g
4 3
This example illustrates a good point. If one knows the matrix of a linear operator in some
ordered basis  and wishes to find the matrix in another ordered basis ', it is often most
convenient to perform the coordinate change using the invertible matrix P; however, it may be
a much simpler task to find the representing matrix by a direct appeal to its definition.
Definition: Let A and B be n×n (square) matrices over the field F. We say that B is similar to A
-1
over F if there is an invertible n×n matrix P over F such that P AP .
According to Theorem 4, we have the following: If V is an n-dimensional vector space over F and
 and ' are two ordered bases for V then for each linear operator T on V the matrix

i
B = T is similar to the matrix A = T . The argument also goes in the other direction. Suppose
'  ' 
A and B are n×n matrices and that B is similar to A. Let V be any n-dimensional space over F and
let be an ordered basis for V. Let T be the linear operator on V which is represented in the basis
-1
 by A. If P AP let ' be the ordered basis for V obtained from  by P, i.e.
,
n
' P .
j ij i
i 1
Then the matrix of T in the ordered basis ' will be B.
Thus the statement that B is similar to A means that on each n-dimensional space over F the
matrices A and B represent the same linear transformation in two (possibly) different ordered
basis.
Note that each n×n matrix A is similar to itself, using P = I; if B is similar to A, then A is similar
1
–1
to B, for B = P AP implies that A P 1 1 BP ; if B is similar to A and C is similar to B, then C is
–1
–1
–1
similar to A, for B = P AP and C = Q BQ imply that C = (PQ) A(PQ). Thus, similarity is an
equivalence relation on the set of n×n matrices over the field F. Also note that the only matrix
similar to the identity matrix I is I itself, and that the only matrix similar to the zero matrix is the
zero matrix itself.
Self Assessment

3. Let T be the linear transformation on R defined by
3
T x 1 ,x x 3 3x 1 x 3 , 2x 1 x 2 , x 1 2x 2 4x 3
,
2
3
(i) What is the matrix of T in the standard ordered basis for R ?
(ii) What is the matrix of T in the ordered basis 1 , 2 , 3 where

1,0,1 , 1,2,1 and 2,1,1
1 2 3
3
2
4. Let T be the linear transformation from R into R defined by
,
T x 1 ,x x 3 x 1 x 2 ,2x 3 – x 1
2
2
3
If B is the standard ordered basis for R and ' is the standard ordered basis for R , what is
the matrix of T relative to the pair , '?

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