From a1cc4a3834d1a2101aba0b74d84c3d7ed0afd1c3 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Wed, 18 Oct 2017 19:19:39 +0200 Subject: [PATCH] update on regressions analysis --- doc/src/Regression/Regression.do.txt | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/doc/src/Regression/Regression.do.txt b/doc/src/Regression/Regression.do.txt index d2a95b773..00cf756e1 100644 --- a/doc/src/Regression/Regression.do.txt +++ b/doc/src/Regression/Regression.do.txt @@ -281,3 +281,16 @@ How can we use the singular value decomposition to find the parameters $\beta_j$ !eblock + + + +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 + +{\displaystyle \mathbf {M} =\mathbf {U} {\boldsymbol {\Sigma }}\mathbf {V} ^{*}} \mathbf {M} =\mathbf {U} {\boldsymbol {\Sigma }}\mathbf {V} ^{*} +where + +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).