Linear Algebra and Its Applications · Problem 18

Positive Definite Matrices — Gilbert Strang

Mathematics · pseudoinverse-least-squares

Problem

What is the minimum-length least-squares solution

x^+ = A^+b

to the following?

Ax = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} C \\ D \\ E \end{bmatrix} = \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}.

You can compute

A^+

, or find the general solution to

A^TA\widehat{x} = A^Tb

and choose the solution that is in the row space of

A

. This problem fits the best plane

C + Dt + Ez

b = 0

and also

b = 2

t = z = 0

(and

b = 2

t = z = 1

). (a) If

A

has independent columns, its left-inverse

(A^TA)^{-1}A^T

A^+

. (b) If

A

has independent rows, its right-inverse

A^T(AA^T)^{-1}

A^+

. In both cases, verify that

x^+ = A^+b

is in the row space and

A^TAx^+ = A^Tb