91Ó°ÊÓ

Find a closest rank-1 approximation to these matrices ( \(L^{2}\) or Frobenius norm): $$ A=\left[\begin{array}{lll} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right] \quad A=\left[\begin{array}{ll} 0 & 3 \\ 2 & 0 \end{array}\right] \quad A=\left[\begin{array}{ll} 2 & 1 \\ 1 & 2 \end{array}\right] $$

If \(a_{11}, \ldots, a_{1 n}\) is the first row of a rank-1 matrix \(A\) and \(a_{11}, \ldots, a_{m 1}\) is the first column, find a formula for \(a_{i j}\). Good to check when \(a_{11}=2, a_{12}=3, a_{21}=4\). When will your formula break down? Then rank 1 is impossible or not unique.

This matrix is singular with rank one. Find three \(\lambda\) 's and three eigenvectors: $$ A=\left[\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right]\left[\begin{array}{lll} 2 & 1 & 2 \end{array}\right]=\left[\begin{array}{lll} 2 & 1 & 2 \\ 4 & 2 & 4 \\ 2 & 1 & 2 \end{array}\right] $$

Suppose two matrices \(A\) and \(B\) have the same column space. (a) Show that their row spaces can be different. (b) Show that the matrices \(C\) (basic columns) can be different. (c) What number will be the same for \(A\) and \(B\) ?