What does it mean for a matrix to be orthogonally diagonalizable?
I'm a little confused as to when a matrix is orthogonally diagonalizable.
I understand that if symmetric, it's always orthogonally diagonalizable, but in what other cases can you orthogonally diagonalize a matrix?
Solution 1:
I assume that by $A$ being orthogonally diagonalizable, you mean that there's an orthogonal matrix U and a diagonal matrix $D$ such that
$$A = UDU^{-1} = UDU^T.$$
A must then be symmetric, since (note that since $D$ is diagonal, $D^T = D$!) $$A^T = \left(UDU^T\right)^T = \left(DU^T\right)^TU^T = UD^TU^T = UDU^T = A \text{.}$$
Solution 2:
A square matrix is said to be orthogonally diagonalizable if there exist an orhtogonal matrix $P$ such that $P^{-1}AP$ is a diagonal matrix.
A square matrix $A$ is orthogonally diagonalizable $\Leftrightarrow$ $A$ is symmetric.
Solution 3:
- For a complex inner product space, a matrix $A$ is orthogonally diagonalizable iff $A A^* = A^* A$.
- For a real inner product space, a matrix $A$ is orthogonally diagonalizable iff $A^T = A$.
Notice that the condition in (2) is more strict than (1) in that (2) $\implies$ (1).
Orthogonal diagonalizability of matrix $A \in \mathbb{F}^{n \times n}$ means there exists an orthonormal basis for $\mathbb{F}^n$ consisting of eigenvectors of $A$.