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Section22Some theorems about the SVD (NM, 23 Oct)

In what follows we will assume each matrix \(A \in \Cmn\) and that \(p=\min(m,n)\text{.}\)

Recall the Frobenius norm of a matrix:

\begin{equation*} \|A\|_F := \big(\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2\big)^{1/2}. \end{equation*}

We concluded with noting that \(A\) can be written as the sum of rank 1 matrices, and idea we will return to in the next class.


  • Suppose that \(D\) is a diagonal matrix; i.e., \(D=\diag(d_{11}, d_{22}, \dots, d_{mm})\text{.}\) Show that each \(d_{ii}\) is an eigenvalue of \(D\text{.}\) What are the corresponding eigenvectors?
  • Suppose that \(L\) is a lower triangular matrix. Show that each \(l_{ii}\) is an eigenvalue of \(L\text{.}\)