Data mining : lecture 8 Edo liberty , adapted from class notes by Yoel Shkolnisky

Edo liberty, adapted from class notes by Yoel Shkolnisky We will see that any matrix A ∈ R m×n can be written as A = U ΣV T such that U ∈ R m×m is unitary, V ∈ R n×n is unitary, and Σ ∈ R m×n is a non-negative real diagonal matrix. Σ(i, i), denoted σ i , are unique. If A the singular values are distinct, then the singular vectors are unique up to a multiplication by z ∈ C with |z| = 1. Remark 0.1. Note the difference in notation from what we saw in class. The matrices V and U are what we denoted by [V ; V ] and [U ; U ] respectively. This makes the proofs a little cleaner and hopefully more easy to follow. Note also that Σ , unlike the matrix we denoted by S, is not square. The non square matrix Σ is still diagonal though, i.e. Σ(i, j) = 0 for all i = j.