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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 ( – Qiaochu Yuan Jun 20 '12 at 1:30
up vote 1 down vote accepted

On an arbitrary field you cannot guarantee the existence of eigenvalues (there might be irreducible polynomials of high degree) nor the singular values (there's not general notion of positivity).

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