The function svd(A) computes the singular values and left and right singular vectors of matrix A.

When the function is called using the form S = svd(A), the returned value of S is a column vector containing the singular values of A.

When the function is called using the form U,S,V := svd(A), the returned value of U is a matrix whose columns are the left singular vectors of A. The returned value of S is a column vector containing the singular values of A.  The returned value of V is a matrix whose columns are the right singular vectors of A.

$\mathrm{with}\left(\mathrm{MTM}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\,1\right]\,\left[2\,4\,2\right]\,\left[2\,8\,1\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 2& 8& 1\end{array}\right]$

$\mathrm{svd}\left(A\right)$

$\left[\begin{array}{c}0\\ \frac{\sqrt{198+2\sqrt{8741}}}{2}\\ \frac{\sqrt{198-2\sqrt{8741}}}{2}\end{array}\right]$

$U,S,V≔\mathrm{svd}\left(A\right)$

$U,S,V≔\left[\begin{array}{ccc}\mathrm{-0.241424496932006}& \mathrm{-0.376449481711859}& \mathrm{-0.894427190999916}\\ \mathrm{-0.482848993864012}& \mathrm{-0.752898963423719}& 0.447213595499958\\ \mathrm{-0.841766631202282}& 0.539841586573655& \mathrm{-5.55111512312578}\times {10}^{\mathrm{-17}}\end{array}\right],\left[\begin{array}{ccc}9.81053809097800& 0& 0\\ 0& 1.65931985026082& 0\\ 0& 0& 3.51436431749812\times {10}^{\mathrm{-16}}\end{array}\right],\left[\begin{array}{ccc}\mathrm{-0.294648032580691}& \mathrm{-0.483670604727495}& \mathrm{-0.824163383692134}\\ \mathrm{-0.932505223882814}& 0.334015093825057& 0.137360563948689\\ \mathrm{-0.208845742900333}& \mathrm{-0.809009680547506}& 0.549442255794756\end{array}\right]$