diagonalizable matrix

1. If a matrix \(A\) is similar (see matrix similarity) to a diagonal matrix, i.e., \(A = PDP^{-1}\), then \(A\) is diagonalizable.
2. An \(n\times n\) matrix is diagonalizable if the sum of its eigenspace dimensions is equal to \(n\). That is, the corresponding eigenvectors form a basis for \(\mathbb{R}^n\).

These two definitions are equivalent.

The columns of \(P\) must be eigenvectors. Imagine taking the \(i\) -th column \(P\), \(p\). Multiply it by \(PDP^{-1}\). When we multiply \(P^{-1}p\) we get \(e_i\), because \(P e_i = p\). When we multiply \(De_i\) we get \(\lambda_i e_i\). When we multiply by \(P\) we get \(\lambda p\).

• real symmetric matrices are always diagonalizable
  • in this case, the matrix of eigenvectors \(Q\) and the matrix where the diagonal entries are the eigenvalues \(D\) give \(A = QDQ^{T}\) (see lecture on the spectral theorem)
  • here, the columns of \(Q\) can be chosen to be orthogonal