We have a matrix $M= \begin{bmatrix} 1 & -1 & 1\\ 2 & 1 & 2 \end{bmatrix}$. My question is how to find a rank one $2 \times 3$ matrix $N$ such that $\|M-N\|_2$ is minimized?
I don't know where to start. I appreciate any hints or solutions.
$M= \begin{bmatrix} 1 & -1 & 1\\ 2 & 1 & 2 \end{bmatrix}$,find a rank one $2 \times 3$ matrix $N$ such that $\|M-N\|_2$ is minimized
2
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linear-algebra
matrices
numerical-methods
norm
numerical-linear-algebra
1 Answers
2
Hint:
Let $$M=\sum_{i=1}^2\sigma_i U_i V_i^T$$ be the SVD decomposition where $\sigma_1 > \sigma_2$.
It is known that the best rank $1$ $2-$norm approximation is $M_1=\sigma_1U_1V_1^T$.
Similar result is known for best rank $k$ $2-$norm approximation or Frobenius norm approximation.