# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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### Tightness of inequalities for various matrix norms

For a general inequality involving matrix norms, does the choice of the norm influence the tightness of the inequality? Eg. In $\|AB\| \leq \|A\| \|B\|$, Does the choice of the norm affect the ...
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### How can I find a norm of a linear transformation $T(x,y) = (ax+cy, bx+dy)$?

Let a linear transformation $T : \mathbb{C}^2 \to \mathbb{C}^2$ s.t $T(x,y) = (ax+cy, bx+dy)$ where $a,b,c,d \in \mathbb{C}$. Now, find the norm of T equipped with ($\mathbb{C}^2$ , $l^1(\{1,2\})$ ...
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### Taking limits of norms of a matrix raised to the nth power:

Given a matrix $$A = \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \end{bmatrix}$$ and a vector $x = \begin{bmatrix}1&0\end{bmatrix}$, compute ...
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### Find a bound for a matrix

Let $A, B \in \mathbb{R}^{n\times n}$ and $I$ be an identity matrix of order $n$. Suppose $$B_k = B_{k-1} + B_{k-1} (I - A B_{k-1}), \quad (k=1,2,\ldots)$$ If $\Vert I - AB_0 \Vert = c < 1$, then ...
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### Relationship between the square of the Frobenius norm and the Frobenius norm of the square

I am looking to understand the following relationship: I have a matrix $A$, whose entries are all bounded by $0 \leq a_{i,j} \leq 1$, and follows the constraint $\|A\|_2 = 1$. Is there anything ...
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### Limits in matrix norm if convergence of integration is guaranteed

the answer of this question probably is very obvious, but I want to make sure this is correct. I have a function $F: \mathbb{R}\rightarrow \mathbb{R}^{n\times n}$ that is continuous and ...
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### Is $\bigl\|\frac{vv^T}{v^Tv}\bigr\|=1$? For any vector $v\in \mathbb{R}^{n}$

I am stuck while showing that $$\biggl\|\frac{vv^T}{v^Tv}\biggr\|=1,$$ where $v\in \mathbb{R}^n$, and $\|.\|$ is a matrix norm. Here is my steps: I used Frobenius norm: A Frobenius matrix ...
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### Condition number - proof

I have a problem with which I've been struggling for a while. It's probably not that difficult, but I seem to be stuck so... here we are. Let A be an invertible lower- or upper-triangular matrix ...
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### Norm preserving Matrix properties

Norm-2 preserving can be done using unitary/orthogonal matrix: $A^*A = I => ||Ax|| = ||x||$ What is the matrix other than identity matrix that can preserve other norms ( norm-1, norm-inf) ?