Dashboard

Now you are in the subtree of Container for Linear Algebra project. 

A matrix is orthogonally diagonalizable if and only if it is normal (The principal axis theorem).

Created over 8 years ago, updated 10 days ago

Principal Axis Theorem (Complex Spectral Theorem). A complex matrix $A$ is unitarily diagonalizable (i.e., there exists a unitary matrix $U$ and diagonal matrix $D$ such that $A = UDU^*$) if and only if $A$ is normal.

Proof ($\Leftarrow$): By Schur decomposition, $A = UTU^*$ with $T$ upper triangular. Since $A$ is normal, $T$ is also normal. A normal upper triangular matrix must be diagonal.

Proof ($\Rightarrow$): If $A = UDU^*$, then $A^* = UD^*U^*$, and $AA^* = UDD^*U^* = UD^*DU^* = A^*A$.

This theorem generalizes the spectral theorem for symmetric/Hermitian matrices and shows that normality is the precise condition for unitary diagonalizability.