[Math] Difference between Jordan decomposition and Eigenvalue decomposition (Spectral decomposition theorem).

Question

I would like to know what are differences between Jordan decomposition and Spectral decomposition theorem (Eigenvalue decomposition). Is main difference that for Eigenvalue decomposition P – matrix with eigenvectors have to be orthonormal?

Answer 1

The spectral decomposition in the "spectral decomoposition theorem" is a special kind of eigenvalue decomposition in which the $P$-matrix is orthogonal (which is to say that the eigenvectors are orthonormal). An eigenvalue decomposition is a special kind of Jordan decomposition in which the "Jordan form" matrix is diagonal.

Every square matrix has a Jordan decomposition. Every diagonalizable matrix has an eigenvalue decomposition. Every normal matrix has a spectral decomposition.