Normal Matrix vs Diagonalizable Matrix

Question

I'm trying to understand the difference between a normal matrix ($A^TA = AA^T$) versus a matrix that can be diagonalized (let's stick with real matrices for this question). In particular, does one of these concepts enclose the other:

• Is a diagonalizable matrix a subset of normal matrices?
• Or, is a normal matrix a subset of diagonalizable matrices?

Perhaps the answer depends on some nuance I haven't appreciated or they are interrelated in a way I haven't come across. I have consulted a number of sources and I don't see a clear answer. Can someone clarify this?

Answer 1

Every normal matrix is (unitarily) diagonalizable (a standard result), but not every diagonalizable matrix is normal.

If $S$ is an invertible matrix that is not unitary (i.e., $S^* \ne S^{-1}$) and $D$ is a diagonal matrix that does not commute with $S$, then $A := SDS^{-1}$ is diagonalizable but not normal.

Answer 2

Take $A=\begin{pmatrix}1&-2\\1&4\end{pmatrix}$ which is diagonalizable but not normal to see why first bullet is false.