Eigendecomposition can be a method to decompose or factor a matrix in different parts, but the problem with Eigendecomposition is that it can work only with the square matrix for the calculation of the inverse matrix, which makes it applicable to a very small set of data.

So there come singular value decomposition which can work with mostly any type of matrix of any shape.

The SVD of matrix $A^{n\ast m}$ is defined by,

where $U^{n\ast n}$ and $V^{m\ast m}$ are orthogonal-matrix and $D^{n\ast m}$ is the diagonal-matrix

The columns of $U$ is called as left singular vector and the columns of $V$ is called as right singular vector.

The columns of $U$ are the eigenvectors of $A{A}^T$ and the columns of V are the eigenvectors of $A^{T}A$.
The diagonal values of $D$ are the eigenvalues of $A{A}^T$ or $A^{T}A$

The eigenvalue or eigenvectors can be calculate using the formula described on eigenvalue-eigenvector.

One small example of SVD can be seen here.

References

• Deep Learning by Ian Goodfellow
• MIT tutorial with example