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Section 25.2 Terminology and notation

similar matrices

a pair of square matrices \(A\) and \(B\) for which there exists an invertible matrix \(P\) satisfying \(B=\inv{P}AP\)

transition matrix

an invertible matrix \(P\) that realizes a similarity relationship \(B=\inv{P}AP\) for similar matrices \(A\) and \(B\)

Note 25.2.1.

We have already used the term transition matrix to mean a change-of-basis matrix (see Section 22.2), and we will justify this double definition of the term when we study similarity in more generality in Chapter 26.

diagonalizable

a square matrix that is similar to a diagonal matrix

The next two definitions apply to an eigenvalue of a square matrix.

algebraic multiplicity

the number of times the eigenvalue is repeated as a root of the characteristic polynomial of the matrix

geometric multiplicity

the dimension of the corresponding eigenspace