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For questions about matrix diagonalization, that is, writing a matrix, a bilinear form or an operator into a "basis" making this one diagonal. This tag is **NOT** for diagonalization arguments from logic and set theory.

A square matrix $A$ is diagonalisable if there is an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. One can view $P$ as a change of basis matrix so that, if $A$ is viewed as the standard matrix of a linear map $T$ from a vector space to itself in some basis, it is equivalent to say there exists an ordered basis such that the standard matrix of $T$ is diagonal.

Not every matrix is diagonalisable over $\mathbb{R}$ (i.e. only allowing real matrices $P$). For example, $$\left[\begin{array}\ 0 & -1\\ 1 & 0\end{array}\right]$$ is not diagonalisable over $\mathbb{R}$.

Diagonalization can be used to compute the powers of a matrix $A$ efficiently, provided the matrix is diagonalizable.

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