# Questions tagged [matrix-norms]

This tag is for questions regarding the Matrix Norm, a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

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### Upper bound of biggest singular value of Kronecker Sum via singular values and traces

I am investigating the biggest singular value of Kronecker Sum. I'd like to understand if my argument is correct. Let $A$, $B$ be square matrices of size $n$. Let $I$ be an identity matrix of size $n$....
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I know that the p-norm of a matrix is defined as: $\displaystyle \|A\|_p = \max_{x\neq0}\frac{\|Ax\|_p}{\|x\|_p}$ However, how can I conclude the below formula from the above formula? $\|A\|_p=\max_{\|... 1 vote 0 answers 42 views ### Bound on minimum perturbation of eigenvalues, based on condition number Problem: This problem comes from a past Ph.D. qualifying exam at my institution. Let$\newcommand{\R}{\mathbb{R}} \newcommand{\l}{\lambda} A \in \R^{n \times n}$be of full rank and diagonalizable as ... 1 vote 1 answer 50 views ### Relation between matrix norms and contractions I'm not so familiar with matrix norms, so I attempted an elementary proof to find conditions on the map$\phi(x) = Ax + b$to be a contraction under the metrics induced by the$L^1$and$L^2$norms. ... 1 vote 0 answers 87 views ### Prove that$\sum_{k=1}^n|\lambda_k(X)-\lambda_k(Y)|\leq\|X-Y\|_1X,Y$are$n\times n$Hermitian matrices.$\lambda_k(X)$denotes the$k$th largest eigenvalue of$X$. Prove that$\sum_{k=1}^n|\lambda_k(X)-\lambda_k(Y)|\leq\|X-Y\|_1$. (Here$\|X-Y\|_1=\sum_{k=1}^n|\...
Setup: Given that I have a random square matrix $X$ where each entry is generated independently and identically. Denote $\|X\|_{\text{op}}$ to be the operator norm of the square matrix. Question: ...