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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1996 • Question 3

Question

Math 6420 • PDE Fall 1996 • Question 3

Let \(A\) be an \(m\times n\) matrix and let the \(n\times m\) matrix \(A'\) denote the Least Square Pseudo-Inverse of \(A\).

  1. State the conditions that define \(A'\).
  2. Show that \(A'\) always exists.
  3. Construct \(A'\) explicitly for the \(2\times 2\) matrix \[ A=\begin {pmatrix} 1 & 1\\ 0.5 & 0.5 \end {pmatrix}. \]

Raw LaTeX

Let \(A\) be an \(m\times n\) matrix and let the \(n\times m\)
matrix \(A'\) denote the Least Square Pseudo-Inverse of \(A\).
\begin{enumerate}
\item[(a)] State the conditions that define \(A'\).
\item[(b)] Show that \(A'\) always exists.
\item[(c)] Construct \(A'\) explicitly for the \(2\times2\) matrix
\[
  A=\begin{pmatrix}
    1 & 1\\
    0.5 & 0.5
  \end{pmatrix}.
\]
\end{enumerate}

Topic Tags

Least Squares and Normal Equations Linear Programming and Optimization Moore-Penrose Pseudoinverse Zero-Stability and Root Conditions