Math 2.0

Linear and multilinear algebra; matrix theory Definition

Diagonalisable matrix

Builds on Similar matricesDiagonal matrix

Definition

Let FF be a field and let AA be an n×nn \times n matrix over FF. The matrix AA is diagonalisable over FF when there is an invertible n×nn \times n matrix PP over FF such that P−1APP^{-1} A P is a diagonal matrix. An endomorphism of a finite-dimensional space is diagonalisable when its matrix in some basis is diagonal.

Remarks

So AA is diagonalisable if and only if it is similar to a diagonal matrix. The diagonal entries of P−1APP^{-1} A P are the eigenvalues of AA, and the columns of PP are eigenvectors. The matrix (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} is not diagonalisable over any field. Over ℝ\R, the rotation matrix (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} is not diagonalisable; over ℂ\C, it is.

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  1. Web https://en.wikipedia.org/wiki/Diagonalizable_matrix, Definition
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DefinesDiagonalisable matrix
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