2
$\begingroup$

I'm studying matrix analysis with Horn and Johnson's book.

I have something trouble while reading the book.

There is lemma 5.6.10 lemma and the following is the proof of that Proof of lemma.

I have trouble in two lines below from the matrix such that 1-norm of $D_t \triangle D_t^{-1}$ is less and equal to $\rho(A)+\epsilon$.

1-norm is defined as the sum of all element in the matrix.

I understood that off-diagonal elements can be bounded by epsilon for large $t$. However, I cannot understand how does the sum of absolute values of eigenvalues will be bounded by spectral radius of $A$.

$\endgroup$
2
$\begingroup$

The $1$-norm of a matrix is usually the max column sum (that is, the max column $\ell_1$-norm). Are you sure that your book defines it as you said?

If it is the max column sum, then the norm will be $|\lambda_j|$ plus off-diagonal elements. The off-diagonal elements are bounded by $\epsilon,$ and the spectral radius will bounded any $|\lambda_j|$, by definition.

$\endgroup$
  • $\begingroup$ Oh I checked that you're right. I was confused since the book talked about the sum of off-diagonal just before that. Thank you so much! :D $\endgroup$ – user562093 May 28 at 12:40
  • $\begingroup$ No problem, I'm glad that it helped! $\endgroup$ – cmk May 28 at 12:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.