# Singular Value Decomposition: Field Type Of Input Matrix

On Wikipedia under Statement of Theorem of SVD it says:

Suppose M is an m×n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization of the form

$$M=U\Sigma V^*$$

where U is an m×m unitary matrix over K, the matrix Σ is an m×n diagonal matrix with nonnegative real numbers on the diagonal, and the n×n unitary matrix V* denotes the conjugate transpose of V. Such a factorization is called the singular value decomposition of M.

Why is it specified that the field K has to be real or complex? Why doesn't it work over a arbitrary field K?

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$K$ needs to have a notion of positivity as well as a notion of conjugacy. I think you can replace it with a real-closed field or its algebraic closure, though (en.wikipedia.org/wiki/Real_closed_field). – Qiaochu Yuan Jun 20 '12 at 1:30