How do you find singular value decomposition of a complex matrix?

Question : How do you find singular value decomposition of a complex matrix?

Solution:

Singular value decomposition:

Let \[A\in M_{m\times n}\left(C\right)\] be any complex matrix. Then we define singular value decomposition of A as \[A=U\mathrm{\Sigma}V^\ast\] where U and V are m\times m and n\times n complex unitary matrices and \[V^\ast\] denotes conjugate transpose of V . \[\mathrm{\Sigma}\] is a rectangular diagonal matrix . It’s diagonal entries are known as singular values of A. We obtain singular values of A by taking square root of non zero eigenvalues of \[A^⁎A\] and \[AA^⁎\] We form columns of matrix U as orthonormal eigenvectors of \[AA^⁎\].and columns of V are orthonormal eigenvectors of \[A^⁎A\]. In this way we obtain the singular value decomposition of a complex matrix.