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 $A$.
Best Answer
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.