Linear Algebra and Its Applications · Problem 21

Positive Definite Matrices — Gilbert Strang

Mathematics · pseudoinverse

Problem

Removing zero rows of

U

leaves

A = \underline{L}\,\underline{U}

, where the

r

columns of

\underline{L}

span the column space of

A

and the

r

rows of

\underline{U}

span the row space. Then

A^+

has the explicit formula

\underline{U}^T(\underline{U}\,\underline{U}^T)^{-1}(\underline{L}^T\underline{L})^{-1}\underline{L}^T

. Why is

A^+b

in the row space with

\underline{U}^T

at the front? Why does

A^TAA^+b = A^Tb

, so that

x^+ = A^+b

satisfies the normal equation as it should?