# 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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### ||A-B|| compared to the difference of their smallest singular values

I came cross a problem: For two matrices $A \in \mathbb{R}^{m \times n}$, $B \in \mathbb{R}^{m \times n}$, if ${\rm rank}\ A = {\rm rank}\ B = r \leq \min \{m,n\}$, then |\sigma_r(A)...
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### Proof of $\|P\|_2=1$ iff $P$ is an orthogonal projector - continuation

I am trying to do the same exercise as in this question: Let $P\in C^{m×m}$ be a non-zero projector. Show that $||P||_2=1$ iff $P$ is an orthogonal projector. I managed to prove everything but the ...
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### Operator norm with respect to the l-2 norm of the matrix that have identical elements

I am interested in a matrix $A \in \mathbb{R}^{m \times n}, m\geq n$, whose elements are identically $a$. Is there a way to relate $||A||$, the operator norm of the matrix induced by vector 2-norm, to ...
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### Norm inequality for eigendecomposition and oblique projectors

Consider a stochastic matrix R which permits an eigendecomposition into oblique projectors, $$R = \sum_{\lambda} \lambda C_{\lambda}$$ I've observed the following 2-norm inequality in a number of $R$ ...
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### Decomposition of a matrix based on vector orthogonality criteria

Given a vector $u\in\mathbb R^{n}$, $u=Wx$, where $x\in\mathbb R^{m}$ and $W$ is a matrix of appropriate dimensions. Let $v\in\mathbb R^{n}$ be a fixed unit vector. I can decompose $u$ into $\perp$ ...
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1 vote
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### "Best" Submultiplicative / Subordinate norm?

I have $y = Ax$ where x, y are vectors and $A$ is a matrix. I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I ...
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### Is there any inequality involving the Frobenius norm and the dimension of matrix?

Let $A$ be a $m \times r$ matrix and $B$ be a $r \times n$ matrix, I wonder if there exists an inequality like the following: $$\left \| AB \right \|_F \leq f(m,r,n)g(A,B) ,$$ or  \left \| AB \...
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### Multidimensional Mean Value Theorem with arbitrary norm

In the question Multivariate Mean Value Theorem Reference was written the following statement for $x,y\in \mathbb{R}^{n}$ ||f(x) - f(y)||_q \leq \sup_{z\in[x,y]}||f'(z)||_{(q,p)}||x-...
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I've found this inequality being used a lot in papers but have been unable to prove it for myself. Consider a matrix polynomial $P(z)=A_{m}z^{m}+...+A_0$ with complex matrix coefficients. Then, for ...