It’s not true: A = full(gallery(‘tridiag’,5,1,2.5,1)) is a counterexample.

However, see Theorem 6 in https://nhigham.com/2021/04/08/what-is-a-diagonally-dominant-matrix/ for a partial result along these lines.

I checked this numerically which holds, but could not find any analytic proof to support this.

Many thanks

]]>I am glad to see your new interesting post, with important references. However, your sentence “structure-exploiting linear system solvers exist that are substantially faster, and potentially more accurate” suggests a new one (useful not only for solving linear systems, but also for the computation of eigenvalues and singular values), a paper which has inspired a lot of work in this field:

-Plamen Koev: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29(3), pp. 731-751 (2007).

https://epubs.siam.org/doi/abs/10.1137/04061903X

Thank you for your wonderful work, and thank you also for giving the readers this freedom to add comments/complements.

]]>Thanks – corrected.

]]>In the example: “…eigenvector and that the other eigenvalue has modulus less than 1” I think you meant to write “less than 2”. ]]>

He taught me a Dfq course while doing my M.SC at Manchester (1979-1980)

He was very nice ,hardworking person

He also was very friendly to me and other overseas students who were experiencing homesick

He and his wife Joan had many comforting conversations with me

My condolences to his family and friends

Big loss to the math community ]]>