1.6. Change of Basis and Similarity 7 V (basis B) W (basis C) V (basis R) W (basis S) ✲ ✲ ❄ ❄ A = [T ]B,C B = [T ]R,S P Q Figure 1.1. Change of basis: BP = QA. Let P be the nonsingular n × n matrix such that [x]R = P [x]B for any x ∈ V. Substitute this into equation (1.2) to get (1.3) [T (x)]S = BP [x]B. Let Q be the nonsingular m × m matrix such that [y]S = Q[y]C for any y ∈ W. Using y = T (x) and equation (1.1), we get (1.4) [T (x)]S = Q[T (x)]C = QA[x]B. Combining (1.3) and (1.4) gives BP [x]B = QA[x]B for any x ∈ V. Hence, BP = QA, and we have B = QAP −1 and A = Q−1BP. In particular, when V = W, B = C, and R = S, we have [T ]B = A and [T ]R = B, with P = Q. Hence, B = QAQ−1. Or we can set S = Q−1 and write B = S−1AS. Definition 1.5. We say two m×n matrices A and B over F are equivalent if there exist nonsingular matrices P and Q, over F, of sizes n × n and m × m, respectively, such that QA = BP . We have shown that two matrices which represent the same linear transforma- tion with respect to different bases are equivalent. The converse also holds i.e., given two equivalent matrices, one can show they may be regarded as two matrix representations for the same linear transformation. Definition 1.6. We say two n × n matrices A and B over F are similar if there exists an n × n nonsingular matrix S such that B = S−1AS. Two n × n matrices are similar if and only if they represent the same linear transformation T : V → V with respect to different bases. For a linear transformation T : Fn → Fn given by an n × n matrix A, so that T (x) = Ax, it will often be convenient to regard T and A as the same object. This is particularly useful when dealing with the issue of similarity, for if B is similar to A, then B may be regarded as another matrix representation of T with respect to a different basis.

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