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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1997 • Question 2

Question

Math 6420 • PDE Fall 1997 • Question 2

Suppose \(A\in \mathbb {R}^{n\times m}\) and \(b\in \mathbb {R}^n\).

  1. Show that the equation \[ A^*Ax=A^*b \] has a solution. Under what conditions on \(A\) is the solution unique?
  2. Show that the eigenvalues of \(A^*A\) are non-negative and that its positive eigenvalues are the same as the positive eigenvalues of \(AA^*\).

Raw LaTeX

Suppose \(A\in\mathbb{R}^{n\times m}\) and
\(b\in\mathbb{R}^n\).
\begin{enumerate}
\item[(a)] Show that the equation
\[
  A^*Ax=A^*b
\]
has a solution. Under what conditions on \(A\) is the solution unique?
\item[(b)] Show that the eigenvalues of \(A^*A\) are non-negative and that its positive
eigenvalues are the same as the positive eigenvalues of \(AA^*\).
\end{enumerate}

Topic Tags

Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization Matrix Rank and Fundamental Linear Algebra Spectral Theory of Operators