# Tagged Questions

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### When does the equality hold for norm equivalence

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these ...
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### How to find x so that $\|A x\| = \|A\| \|x\|$ holds

The subbordinance property of matrix-vector multiplication states that $\|A x\| \le \|A\| \|x\|$ where $\|x\|$ is the norm of vector $x$ and $\|A\|$ is the induced norm of matrix $A$. Many textbooks ...
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### Physical meaning of norm of a matrix

I know norm of a vector is a length of a vector from origin. So what is the motivation behind defining the norm of the matrix? What is the physical meaning of norm of a matrix? Any help is ...
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### $\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always bounded

Given the matrix $A= (a_{i,j}) \in M_{n,n}$ $||A||=\sup\limits_{x\in X}\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ where $||$ . $|| _n$ is $R^N$ norm $\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always ...
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### Matrix norm can assume infinite values?

Given a real-valued matrix $A=a_{i,j} \in M_{n,n}$ When: $||A||_2= \sqrt {\sum_{i=0}^n a^2_{i,j}} < + \infty$ ? Why?
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### Prove or disprove the existence of a length preserving non-normal matrix

Prove or disprove: There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a normal matrix There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a unitary ...
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### Matrix-Vector Product

Suppose I have the expression $\lVert\mathbf B \cdot\hat n\lVert=1$, where $\mathbf B$ is a matrix and $\hat n$ is a unit vector (both can have any dimensions, as long as they are compatible). What ...
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According to [1], [2] and other related publications, the following holds for any matrix $X$: $$\| ... 1answer 30 views ### Restrictions on a Matrix-Vector product Suppose I have a m\times n matrix \mathbf M, and a unit vector \hat v, of dimension n. What restrictions do I need to apply to \mathbf M so that \lVert \mathbf M\cdot \hat v\lVert \leq 1 ... 2answers 44 views ### Computing an induced matrix norm Assume I have a n \times n matrix and a norm defined as \|A\| = \max \limits_{x \not = 0}\frac{\|Ax\|}{\|x\|}, where \|x\| = \sqrt{\sum x_i^2}. How can I compute this norm? 1answer 27 views ### Norm of a Matrix-vector product Suppose I have vector \vec x \in \mathbb R^n and matrix \mathbf M of dimension m\times n. Is there an alternative expression for \lVert \mathbf M \cdot \vec x \lVert that includes \lVert \vec ... 2answers 104 views ### What is the Hessian of Frobenius norm As we know that every norm is convex, and if a function is convex w.r.t. the input variable, then corresponding Hessian should be positive semidefinite. When I try to find the Hessian of Frobenius ... 1answer 79 views ### Inequality of Frobenius norm for skew matrices Let A be a complex skew-symmetric n \times n matrix, that is, A^T = -A. Denote by \|\cdot\|_F the Frobenius norm, that is, \|B\|_F^2 = \text{trace}(B^*B). I would like to prove that$$ ...
$\left | \left | A \right | \right |_{2} :=\left ( \sum_{i,j=1}^{n} a_{ij}^{2} \right )^\frac{1}{2}$ for $A\in \mathbb{R}^{n\times n}$ Show this: If A is a simetric matrix then \left | \left | A ...