Eigendecomposition#
Definition: Diagonalizability
A square matrix \(A \in F^{n \times n}\) is diagonalizable if the endomorphism \(f: F^n \to F^n\) whose matrix representation with respect to the standard basis of \(F^n\) is \(A\) is diagonalizable.
Theorem: Diagonalizability
A square matrix \(A \in F^{n \times n}\) is diagonalizable if and only if there exists an invertible matrix \(P \in F^{n \times n}\) such that
\[D = P^{-1} A P\]
is a diagonal matrix.
In this case, if \(\lambda_1, \dotsc, \lambda_n\) are the (potentially not distinct) eigenvalues of \(A\), then the \(j\)-th entry on the diagonal of \(D\) is \(\lambda_j\) and the \(j\)-th column of \(P\) is an eigenvector of \(A\) associated with \(\lambda_j\).
Proof
TODO