update on regressions analysis

doc/src/Regression/Regression.do.txt

@@ -281,3 +281,16 @@ How can we use the singular value decomposition to find the parameters $\beta_j$
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+Suppose M is a 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, called a singular value decomposition of M, of the form
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+{\displaystyle \mathbf {M} =\mathbf {U} {\boldsymbol {\Sigma }}\mathbf {V} ^{*}} \mathbf {M} =\mathbf {U} {\boldsymbol {\Sigma }}\mathbf {V} ^{*}
+where
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+U is an m × m unitary matrix (if K = {\displaystyle \mathbb {R} } \mathbb {R} , unitary matrices are orthogonal matrices),
+Σ is a diagonal m × n matrix with non-negative real numbers on the diagonal,
+V is an n × n unitary matrix over K, and
+V∗ is the conjugate transpose of V.
+The diagonal entries σi of Σ are known as the singular values of M. A common convention is to list the singular values in descending order. In this case, the diagonal matrix, Σ, is uniquely determined by M (though not the matrices U and V, see below).