1. 4.3 Similarity
If L is a linear operator on an n-dimensional vector space V, the matrix representation of L will depend on the ordered basis chosen for V. By using different bases, it is possible to represent L by different n×n matrices. In this section, we consider different matrix representations of linear operators and characterize the relationship between matrices representing the same linear operator. Let us begin by considering an example in 𝑅 2 . Let L be the linear transformation mapping 𝑅 2 into itself defined by

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3. 4.3 Similarity
c1 – c2 =-2
c1 +c2=0
c1=-c2

4. 4.3 Similarity
Similar Matrices Let A and B be n×n matrices. We say that A is similar to B if there is an invertible n×n matrix P such that P−1AP=B. If A is similar to B, we write A∼B The matrix P depends on A and B. It is not unique for a given pair of similar matrices A and B.

5. 4.3 Similarity EXAMPLE Let be the linear operator defined by
and let be the basis, where
Find [T ] 𝐵
From the given formula for T,

6. where P is the transition matrix from 𝐵 ˊ to B
4.3 Similarity
where P is the transition matrix from 𝐵 ˊ to B
Therefore, Consequently, EXAMPLE Let be defined by Find the matrix of T with respect to the standard basis for 𝑅 2 ; then use above Theorem to find the matrix of T with respect to the basis , where .

7. 4.3 Similarity
Solution To find [T ] 𝐵 , we will need to find the transition matrix
By inspection so

8. 4.3 Similarity Thus the transition matrix from 𝐵 ˊ to B is
If A and B are square matrices, we say that B is similar to A if there is an invertible matrix
P such that

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EXAMPLE 2 Let L be the linear operator mapping R3 into R3 defined by L (x) = Ax, where
Thus the matrix A represents L with respect to {e1, e2, e3}. Find the matrix representing L with respect to {y1, y2, y3}, where

13. 4.3 Similarity
Solution
Thus, the matrix representing L with respect to {y1, y2, y3} is
We could have found D by using the transition matrix Y = (y1, y2, y3) and computing

14. 4.3 Similarity
Example