3A.10 Matrix Transformations

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3A.10.1 Similarity Transformation

Consider a square matrix, A; and a nonsingular square matrix, T: Then, the matrix obtained according to

B ¼ T21AT

is the similarity transformation of A by T: The transformed matrix B has the same eigenvalues as the

original matrix A: Also, A and B are said to be similar.

3A.10.2 Orthogonal Transformation

Consider a square matrix A and another square matrix T: Then, the matrix obtained according to

B ¼ TTAT

is the orthogonal transformation of A by T:

If T21 ¼ TT then the matrix T is said to be an orthogonal matrix. In this case, the similarity

transformation and the orthogonal transformation become identical.

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