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[tex] A = U \Sigma V[/tex]

into the product of an mxm matrix U with orthonormal columns, the mxn matrix ∑ with ∑ = diag([tex]\sqrt{\lambda_i}[/tex]), and the nxn matrix V with orthonormal columns.

In case the definition doesn't provide much help, the V has the orthonormalized eigenvectors of (A^T)A, and U has the orthonormalized eigenvectors of A(A^T).

Do the first r columns of U span A, i.e. do the first r columns of U form a basis for the range of A? Similarly, will the first r columns of V form a basis for the corng of A?

Really what I am trying to determine is if C is a 3x3 matrix with eigenvalues of 0, 1, and 2, if the eigenvalues of C^T C can be determined with the eigenvalues of C.