Is a matrix diagonalizable if and only if it has an eigendecomposition?
If not, can you give an example of a diagonalizable matrix which doesn't have an eigendecomposition, or vice-versa?
2026-03-29 10:07:53.1774778873
Eigendecomposition and Diagonalization of a matrix.
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Yes. If $M$ is a $n\times n$ matrix and if there are vector $v_1,\ldots,v_n$ each of which is an eigenvector of $M$ and such that $\{v_1,\ldots,v_n\}$ is a basis of $M$, if $P$ is the $n\times n$ matrix whose columns are the $v_k$'s, then $P^{-1}MP$ is a diagonal matrix.
And if $M$ is diagonalisable and $P$ is an invertible matrix such that $P^{-1}MP$ is a diagonal matrix, then the columns of $P$ are a basis of eigenvectors of $M$.