2.8 Similarity Transformations

If A and B are n × n matrices, A is similar to B if there exists an invertible matrix P such that

𝐁=𝐏𝐀𝐏-𝟏\mathbf{B=PA{P}^{-1}} (11)

Every matrix is similar to itself with P = I. The only similarity transformation that holds for the identity matrix or the zero matrix is this trivial one.

Similarity is a symmetric relation. If A\simB, then B\simA. Therefore, premultiplying Equation 11 by P-1 and postmultiplying it by P yields

𝐀=𝐏-𝟏𝐁𝐏\mathbf{A={P}^{-1}BP} (12)

Similarity is also a transitive relation. If A\simB and B\simC, then A\simC.

Since similarity is reflexive, symmetric, and transitive, it is an equivalence relation. A common example of a similarity transformation in linear algebra is changing the basis of a vector space.