Frame diagonalization of matrices

This paper introduces a concept of diagonalization that uses not a basis of eigenvectors, but a frame. A frame is a generalization of a basis which is used in a number of signal and image processing applications. We first investigate the properties of frame diagonalization, drawing parallels with those of basis diagonalization. We then describe several methods of constructing frames for frame diagonalization. In particular, we prove the existence of a universal diagonalizer for each n∈N that simultaneously diagonalizes all matrices in Mn(C), and create a method of frame diagonalization that works for any matrix in Mn(C), uses at most ⌊3n/2⌋ frame vectors and retains information about the eigenvalues of the matrix.