Square matrices and are similar when for some invertible matrix . They represent the same linear operator in different bases and therefore have the same Jordan normal form.
A similarity transformation sends a square matrix to for an invertible matrix . It changes the basis of a linear operator and preserves its characteristic polynomial and eigenvalues.
If a real matrix has a positive diagonal matrix such that is a symmetric matrix, then is a symmetric matrix related to by a similarity transformation: gives . Thus is a diagonalizable matrix with real eigenvalues. If for all real vectors , its eigenvalues are nonpositive. This converts symmetry in a weighted inner product into Euclidean symmetry.
Every unital algebra automorphism has the form for some invertible . One proof transports the matrix units through , chooses compatible bases for their rank-one images, and reconstructs the common similarity transformation.
If real matrices satisfy for an invertible complex , then , so both real matrices intertwine and . Since is a nonzero real polynomial, some real makes invertible and supplies a real similarity.
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Matrix similarity is an important concept in linear algebra that describes a relationship between two square matrices. Two matrices \( A \) and \( B \) are said to be similar if there exists an invertible matrix \( P \) such that: \[ B = P^{-1} A P \] In this expression: - \( A \) is the original matrix. - \( B \) is the matrix that is similar to \( A \).
