An extension of spectral decomposition on a square matrix is called the singular value decomposition where any matrix can be decomposed. The trick is to diagonalize the matrix A A^T and A^T A, in this manner we can find the left and right singular values and have an equation which is very similar to the spectral decomposition. It has a very wide range of applicibility--most of which is also highly valuable.
There's also the Moore-Penrose pseudoinverse which also has plenty of applicaitons and works on non-square matrices.
There's also the Moore-Penrose pseudoinverse which also has plenty of applicaitons and works on non-square matrices.