Any normal operator on a vector space is diagonal with respect to some orthonormal basis for .
Conversely, any diagonalizable operator is normal.
Now we have
Thus, if is normal then where and thus is diagonalizable wrt orthonormal basis for . Similarly, is also diagonalizable wrt some orthonormal basis for .
Thus, is diagonalizable for orthonormal basis of the entire vector space.
Polar Value Decomposition
Let be a matrix on vector space . Then there exists unitary and positive operators and such that, where the unique positive operators shall satisfy the equations and .
Moreover, if is invertible then is unique.
Singular Value Decomposition
Let be a square matrix. Then there exist unitary matrices and , and a diagonal matrix with non-negative entries such that .
The diagonal elements of are called the singular values of .
From polar value decomposition we have, .