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 nn 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 . Let L be the linear transformation mapping into itself defined by
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c1 c2 =-2 c1 +c2=0 c1=-c2 3 4.3 Similarity Similar Matrices Let A and B be nn matrices. We say that A is
similar to B if there is an invertible nn matrix P such that P1AP=B. If A is similar to B, we write AB The matrix P depends on A and B. It is not unique for a given pair of similar matrices A and B. 4 4.3 Similarity EXAMPLE
Let and let be the linear operator defined by be the basis, where (a)Find [T
From the given formula for T, 5 4.3 Similarity where P is the transition matrix from to B Therefore,
Consequently, EXAMPLE Let be defined by Find the matrix of T with respect to the standard basis above Theorem to find the matrix of T with respect to the basis , .
for ; then use where 6 4.3 Similarity Solution
To find [T we will need to find the transition matrix By inspection so 7 4.3 Similarity Thus the transition matrix from 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 8 4.3 Similarity 9
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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 12 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 13
4.3 Similarity Example 14