A problem from Gilbert Strang's Linear text, after the section on the singular value decomposition:
Suppose $A$ is $2\times 2$ and invertible (with $\sigma_1 > \sigma_2 > 0$). Change $A$ by as small a matrix as possible to produce a singular matrix $A_0$. Hint: $U$ and $V$ don't change. Use $$A = \left [ \begin {matrix} u_1 & u_2\end {matrix}\right ] \left [ \begin {matrix} \sigma_1 & \\ & \sigma_2 \end {matrix}\right ] \left [ \begin {matrix} v_1 & v_2\end {matrix}\right ]^T.$$
I think we can change $A$ by
$$\left [ \begin {matrix} u_1 & u_2\end {matrix}\right ] \left [ \begin {matrix} 0 & \\ & -\sigma_2 \end {matrix}\right ] \left [ \begin {matrix} v_1 & v_2\end {matrix}\right ]^T.$$
But how can we show this is the smallest possible alteration of $A$ to get a singular matrix?
I think Strang is expecting a somewhat loose answer here, since he's not yet introduced the matrix norm, so it's not clear what he means by "as small as possible." If we were to attempt it in a more rigorous way, I think we'd like two results:
- A result relating singular values to addition and subtraction, something like $\sigma_n(A) + \sigma_n(B) \geq \sigma_n(A \pm B)\geq \sigma_n(A) - \sigma_1(B)$, where $\sigma_1, \dotsc , \sigma_n$ are arranged in descending order.
- A result relating singular values to matrix norm. I believe the largest singular value is the value of the 2-norm, but how can we relate this to the operator norm?