Why is A diagonalizable but not orthogonally diagonalizable?

Question

Could you give me an example of a matrix A that is diagonalizable but not orthogonally diagonalizable? and why? I'm trying to understand the difference between the two.

I think I got it. Thank you :)

Answer 1

Pick any diagonalizable matrix with simple eigenvalues whose corresponding eigenvectors are not orthogonal.

For example, there is a unique $2\times 2$ matrix which has $(1,1)$ and $(1,0)$ as eigenvectors for the eigenvalues $1$ and $2$. Find it.

Answer 2

Assuming a real matrix. Let's write the eigen-equation in matrix form $U=[u_1,u_2...u_n],\Lambda=diag(\lambda_1,\lambda_2,...,\lambda_n)$ $$ MU=U\Lambda $$ Since $M$ is diagonalizable, $U$ is a full set of $n$ linear independent vectors. then $U$ is invertible. $$ M=U\Lambda U^{-1} $$

If a matrix is orthogonal diagonalizable, then $U^T=U^{-1}$ then $M=U\Lambda U^T$, then $M=M^T$. We arrive at the well-known fact that orthogonal diagonalizable real matrices are symmetric (and vise-versa, real symmetric matrix has only real eigenvalues and is orthogonal diagonalizable.).

So only non-symmetric matrices could be diagonalizable but not orthogonal diagonalizable. If we are in real matrix domain, then only these matrices can have complex eigenvalues.

For a simple example (up to numerical precision), a simple non-symmetric matrix is not orthogonally diagonalizable ($u_1^Tu_2\neq 0$). $$ M=\begin{bmatrix} 0 & 2\\ 1 & 0 \end{bmatrix}\\ U= \begin{bmatrix} 0.8165 & -0.8165\\ 0.5774 & 0.5774\\ \end{bmatrix},\; \Lambda = \begin{bmatrix} 2.4142 & 0\\ 0 &-0.4142 \end{bmatrix} $$

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Geometrically, what does this mean? for symmetric matrix (orthogonal diagonalizable), when applying it to a vector $Mv$, it's outcome can be understood as scaling the projections onto each eigenspace by each eigenvalue independently $$ Mv = U\Lambda U^Tv=\lambda_i (v^Tu_i)u_i $$ In your mind you can think of $M$ in this manner: rotate to the eigenbasis, scale each axis and rotate back. credit to this

For non-symmetric / not orthogonal diagonalizable, you cannot understand it as such. The eigenvectors do not form an orthogonal basis, and the effect of matrix on vectors involves matrix inversion (basis change to the non-orthogonal eigen basis). $$ Mv = U\Lambda U^{-1}v\neq\lambda_i (v^Tu_i)u_i $$ Something like this will be the effect of a non-symmetric matrix.