2

Looking clues for this problem.

Find all the matrices such that $\kappa(A) = 1$

We define $\kappa(A) = \|A\|\,\|A^{-1}\|$. If I'm looking matrices such that $\kappa(A) = 1$, I was thinking in this:

1) If $A = I$, the identity matrix, then $\kappa(A) = 1$.

2) If A is orthogonal, then the columns of $A$ form an orthonormal basis (orthonormal if we consider the euclidian norm) of $\mathbb{R}^{n \times n}$, then $\kappa(A) = 1$.

I'm not sure how to consider the norms $\| \cdot \|_1,\| \cdot \|_{\infty}$, because case (2) is valid whne we consider the euclidian norm.

Thanks for all your help!

Alexei0709
  • 1,184
  • Is the problem asking only about the condition number with respect to the $2$-norm? I think the SVD gives a good way to look at it. – littleO Oct 20 '15 at 00:28
  • No, the problem is asking about the condition number with respect to the $1$-norm and to the $\infty$-norm – Alexei0709 Oct 20 '15 at 02:09
  • I am not sure this is helpful but here is my observation (for orthogonal matrices and norm $|\cdot|1$ and $|\cdot|\infty$). So, norm $|\cdot|{\infty}$ equals to the maximum over rows, of sum of (absolute value) elements in a row, norm $|\cdot|{1}$ equals to the maximum over columns of sum of (absolute values) of elements in a column. If the matrix $A$ is orthogonal then $A^{-1} = A^T$ so the rows and columns are interchanged. Therefor, with either of the two specific norms, we obtain $\kappa(A) = |A|{\infty}|A|_1$ and you must require $|A|{\infty} = |A|_1$ – them Aug 20 '16 at 09:20

0 Answers0