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Questions tagged [matrix-norms]

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18 views

Jacobi iteration for linear systems. Proof of inequality.

I need help with this problem : Let $A =(a_{ij}) \in \mathcal M_n(\mathbb R)$ be an invertible matrix with $ |a_{ii}| \not=0$ for all $i$. Then A can be decomposed into a diagonal component D, and ...
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1answer
28 views

About subgradient of matrix norm

I am reading Characterization of the Subdifferential of Some Matrix Norms by G.A. Watson. And in the first page the subgradient of $\|A\|$ is defined:$$\partial\|A\| := \{G\in \mathbb{R}^{m \times n}:\...
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1answer
26 views

Proving the infinity norm is equal to the maximum value of the vector

We know that . I am trying to figure out how to prove when p goes to infinity then the norm represent the maximum value of the vector
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0answers
25 views

Proof of infinity matrix norm

Given the $l_{\infty}$ matrix norm for $A{\in}{\Bbb{R}}^{mxn}$ is defined as: $\|A\|_{\infty} =\max_{1 \leq i \leq n}\|a^{i}\|_{1}$ (where $a^{i}$ is the i$^{th}$) row in matrix A), Show that: $\|A\|...
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0answers
31 views

Matrix norm for two matrices simultaneously close to spectral radius

Suppose $A$ and $B$ have the same spectral radius $\rho$. We can find a norm $|| \cdot ||_A $ s.t. $||A||_A - \epsilon < \rho$. We can likewise find a another norm s.t. $||B||_B - \epsilon < \...
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1answer
23 views

Can an upperbound constraint on the squared Frobenius norm of a matrix be expressed as a linear matrix inequality?

Is it possible to express an inequality constraint on the squared Frobenius norm of a matrix $X$: $$\|X\|_F^2 = \mathop{tr}( X^T X ) \le t$$ as a linear matrix inequality? I want to say that it's: ...
0
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1answer
33 views

Prove that $cond(A)\ge \frac{||A||}{||A-B||}$ for any induced matrix norm

Prove that for any induced matrix norm: $cond(A)\ge \frac{\left\lVert A \right\rVert}{\left\lVert A-B \right\rVert}$ Where $A$ is an invertible matrix, and $B$ is a singular matrix. The condition ...
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1answer
34 views

1 norm $\|\|_1$, of non square matrix

Does $1$ norm exist for non-square matrices? By $1$ norm I mean $d (x,y)=\sum_{i=1}^{n} |x^i-y^i|, x=(x_1,\dots, x_n), y=(y_1,\dots, y_n)$ Suppose $A$ is $m\times n, (m\ne n)$ matrix what can we ...
2
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1answer
109 views

$L^2$ norm of a matrix: Is this statement true?

I am following Nocedal and Wright's Numerical Optimization book for self study. In the Appendix section of the book, the following matrix norms are defined: They defined the $l2$ norm of the matrix $...
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0answers
24 views

Sum of $k$ smallest singular values

The $k$th Ky Fan norm $\lVert\cdot\rVert_{(k)}$ is defined as the sum of the $k$ largest singular values. Furthermore, for an $m\times n$ matrix $A$ $$ \lVert A\rVert_{(k)} = \max_{UU^*=VV^*=I_k}|\...
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1answer
30 views

$A^t\to 0$ when its row sum is strictly less than one?

$A_{n\times n}$ is a matrix having each row sum $<1$ and its largest eigenvalue is also $<1$. I need to show $A^t\to 0,\text{ i.e } a^t_{ij}\to 0\forall i,j\text{ as } t\to\infty$ given that $0&...
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1answer
21 views

Revisit “Inequality between Frobenius and nuclear norm”

I am reading the following question: Inequality between Frobenius and nuclear norm I know $$\|X\|_* = \sum_{i=1}^r \sigma_i(X)$$and $$\|X\|_F = \bigg(\sum_{i=1}^r \sigma^2_i(X)\bigg)^{1/2}$$ I try ...
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2answers
19 views

Upper bound of the norm of a matrix difference using an absolutely converging geometric series and Neumann's theorem

I am having trouble proving the following statement: Let $\mathbf{A}\in\mathbb{C}^{n\times n}$ be a square matrix such that $\|\mathbf{A}\|<1$, for some induced norm $\|.\|$. Then, $\|(\mathbf{...
2
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1answer
65 views

norm of difference of two matrices.

For any two symmetric $n\times n$ matrices $A$ and $B$, let their eigenvalues be ordered from largest to smallest. How to prove that for eigenvalues $$|\lambda_k^A-\lambda_k^B| \leq \|A-B\| \ \text{...
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1answer
53 views

Does a matrix of minimum norm in an affine subspace of $M_n(\mathbb R)$ have minimum spectral radius?

Let $\mathcal U \in M_n(\mathbb R)$ be a subspace defined by declaring certain entries to be $0$. More precisely, let $\Lambda, \Theta \subset \{1, \dots, n\}$, $\mathcal U$ is defined as \begin{align*...
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1answer
27 views

Does $\lVert B^TA^{-1}B\rVert_2<1$ imply that $\left\lVert\begin{bmatrix}0&-A^{-1}B\\0&-B^TA^{-1}B\end{bmatrix}\right\rVert_2<1$?

Does $\left\lVert B^{\operatorname{T}}A^{-1}B\right\rVert_2<1$ imply that $\left\lVert\begin{bmatrix}0&-A^{-1}B\\0&-B^{\operatorname{T}}A^{-1}B\end{bmatrix}\right\rVert_2<1$, for $A,B\in\...
3
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1answer
188 views

When does $\Vert AB \Vert = \Vert A \Vert \Vert B \Vert$?

Motivation If $a$ and $b$ are vector, then thinking simply vector 2 norm, $\Vert a \cdot b\Vert = \Vert b\Vert \Vert a\Vert \cos(a,b) $, we know the difference is simply a ratio between the angle of $...
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2answers
45 views

Prove eigenvalues of a symmetric matrix are in a certain interval

I am given a matrix $A=\begin{bmatrix}1&2&0\\0&1&2\\0&0&1\end{bmatrix}$. I am asked to compute $A^tA=\begin{bmatrix}1&2&0\\2&5&2\\0&2&5\end{bmatrix}$ ...
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1answer
118 views

Frobenius and operator-2 norm

I have been studying about norms and for a given matrix A, I haven't been able to understand the difference between Frobenius norm $||A||_F$ and operator-2 norm $|||A|||_2$. Can someone help me ...
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1answer
23 views

Inequality for the norm of a matrix and the norm of its columns and rows.

Let $\boldsymbol{M}$ be an $n \times m$ matrix. Let $\boldsymbol{M}_{i,:}$ be the $i^{th}$ row of $\boldsymbol{M}$ and $\boldsymbol{M}_{:,j}$ by the $j^{th}$ column. Is it the case that: $$||\...
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1answer
25 views

Prove a matrix norm inequality

Given an induced matrix norm $||\cdot||$ such that $\exists_{\varepsilon >0} \forall_{x\in\mathbb{R} ^n} ||Ax||\ge\varepsilon ||x||$ prove that $||A^{-1}||\leq\frac1\varepsilon$. I figured out ...
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2answers
69 views

Frobenius Norm Inequality with SVD

Let $A\in \mathbb{R}^{m\times n}$ and $x\in \mathbb{R}^n$ a column vector. I want to prove that $$||Ax||_2 \leq||A||_F||x||_2$$ using SVD where $||\cdot||_2$ is the euclidean norm and $||\cdot||_F$ ...
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1answer
74 views

Number of Solutions to the Problem: Minimize $|| A - Afc^T / c^T f ||^2$ such that $\sum_i c_i = 1$

Let $A$ be a $M$ by $N$ matrix, $f$ is a column matrix with $N$ elements, $c$ is a column vector with $N$ elements that I need to solve for, and $|| \cdot ||$ is the Frobenius norm of the matrix. I ...
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1answer
167 views

Frobenius Norm Inequality; Spectral Radius is smaller than Frobenius Norm

Let $A \in \mathbb{R}^{n \times m}$ and $x \in \mathbb{R}^n$. Prove the following inequality. $\left\lVert \cdot \right\rVert_F$ denotes the Frobenius norm and $\left\lVert \cdot \right\rVert_2$ ...
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0answers
21 views

Show inequality for Euclidean norm on SPD matrix and identity matrix

Let A be a symmetric, positive definite matrix. Show that in the Euclidean norm $$||I-\frac{1}{\tau}A||_{2}<1$$ implies that $0<\tau<2||A||^{-1}_{2}$ for $\tau$ a scalar.
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1answer
24 views

Matrix norm compatible with Euclidean norm that is tighter than Frobenius

Define the matrix-norm $\|M\|=\left(\sum_{j_1,j_2=1}^n\left|\sum_{i=1}^nM_{ij_1}M_{ij_2}^*\right|^2\right)^{1/4}$ This is smaller than the Frobenius norm and it compatible with the vector 2-norm. Q:...
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0answers
18 views

Deriving an inequality related to an induced matrix norm

Suppose we have the induced matrix norm for an arbitrary $n\times n$ matrix $A$ given by, \begin{equation*} |||A|||_{\infty,h} = \max_{1\leq i \leq N}\left(\sum_{j=1}^N |A_{ij}|\right). \end{equation*...
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2answers
30 views

It can happen that the norm 1 of a matrix and the infinite norm are different?

I have practiced some exercises with these two norms and in all of them I had the same result, until I tried with $\begin{pmatrix} 5 & -3 & 2 \\ 4 & 8 & -4 \\ 2 & 6 & -1 \\ \...
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0answers
21 views

Bound on eigenvalues of matrices of the form $XDX^T$

I encountered this form of a Matrix while analyzing Logistic Regression (It's the Hessian). Let $H = XDX^T$, where $D$ is a positive definite diagonal matrix with maximum diagonal entry as some $c$ ...
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0answers
49 views

Do the two linear operators on $M_n(\mathbb R)$ have the same induced norm on the subspace of symmetric matrices?

Let $A \in M_n(\mathbb R)$ be fixed with spectral radius $\rho(A) < 1$. Then $T_1, T_2$ are two well-defined linear operators on $M_n(\mathbb R)$ given by \begin{align*} T_1(X) = \sum_{k=0}^{\infty}...
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1answer
47 views

Check if the function $h(A) = \lim\limits_{n \to +\infty} \frac{\text{tr}((A^A)^{n+1})}{\text{tr}((A^A)^{n})}$ is a valid matrix norm

For a given function $$h(A) = \lim\limits_{n \to +\infty} \dfrac{\text{tr}((A^A)^{n+1})}{\text{tr}((A^A)^{n})}$$ we have to check if it's a valid matrix norm. I know that $A^A$ is defined as $$A^A ...
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0answers
130 views

Matrix Norm and Euclidean Norm

A vector, say for example $ a = \begin{bmatrix} 1 \\ 2\end{bmatrix}$, can be regarded not only as a vector but also as a 2 x 1 matrix. If we interpret it as a vector we can compute the $\ell_p$ norm ...
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0answers
51 views

2 norm of a Vandermonde matrix

Let the matrix $V$ be Vandermonde of size $n \times n$: $$V(x_1, \dotsc, x_n) = \begin{bmatrix} 1 & x_1 & x_1^2 & \dotsb & x_1^{n-1} \\ 1 & x_2 & \ddots & & \vdots\\ \...
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1answer
92 views

Distance between the spans of two matrices

We have two matrices of the same dimension, $V_1$ and $V_2$. We take $$ P_i=V_i(V_i^tV_i)^{-1}V_i^t. $$ Then we take the Frobenius norm (Hilbert Schmidt norm) of $P_1-P_2$. Why does this not change ...
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2answers
109 views

Computing the spectral norm of a projection matrix

I was reading a paper in which there was an argument as trivial, but could not make myself sure about it. It is said that given a full row-rank matrix $A$, the norm (probably $\ell_2$-induced matrix ...
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0answers
71 views

Norm of sum of shifted outer products

Consider two vectors $u, v \in \mathbb{R}^m$ satisfying $\| u \|_2 = \| v \|_2 = 1$ and $\#\mathrm{supp}(v) \leq n < m$, where $\mathrm{supp}$ denotes the support of the vector (i.e. locations of ...
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1answer
68 views

Bounding the matrix $2$-norm of a Frobenius matrix

This question arises as a result of a close reading of a proof in the following paper: Buckwar, E. & Winkler, R. Multistep methods for SDEs and their application to problems with small noise. ...
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1answer
116 views

Upper bound of the spectral norm of a matrix power

Let $A\in\mathbb{C}^{n\times n}$ be a complex valued square matrix which can be written as $A=PUP$ in which $P$ is a projector and $U$ is a unitary matrix. The interesting case is $P$ and $U$ do not ...
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1answer
55 views

How to Compute Norm of Matrix?

While looking into the camera rotation dataset, I had found a $3\times3$ rotation matrix $R$ which has very slight change from original : $$R=\begin{bmatrix} 0.99995284&-0.01584106&-0.01266612\...
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1answer
71 views

Weighted Inner Product vs Norm Relation

Let $A$ be a matrix with $\lambda_1>0$. Then we know that, \begin{equation} \frac{||x^TAy||}{|||x|||y||} \leq \lambda_1. \end{equation} But is there any lower bound known for the quantity $\frac{||...
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0answers
20 views

Bounds for Spectral Noem

Let $A\in M_n$ and let $\epsilon>0$ be given. Show that there is a nonsingular matrix $C=C(\epsilon)\in M_n$ such that $\rho(A)<|||CAC^{-1}|||<\rho(A)+\epsilon$. I know we need to use Schur'...
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1answer
308 views

Operator norm (induced $2$-norm) of a Kronecker tensor

Let $A \in \mathcal M(n \times n; \mathbb R)$ with $\rho(A) < 1$. Then we know $I \otimes I - A^T \otimes A^T$ is invertible where $\otimes$ denotes kronecker product. Let $\text{vec}$ denote the ...
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1answer
95 views

Upper bound this family of matrices in induced $2$-norm

Let $\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le M \text{ and } \rho(A) < 1\}$ where $M \ge 1$ is some fixed constant and $\|\cdot\|_2$ denotes the induced $2$-norm. Is it ...
2
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1answer
41 views

Is it possible to upper bound this family of matrices in operator norm?

Let $\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le \|A_0\|_2\}$ where $A_0 $ is some fixed matrix and $\|\cdot\|_2$ denotes the induced $2$-norm. We also have for every $A \in \...
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1answer
189 views

Spectral norm minimization

I was reading the use of semidefinite programs to formulate the matrix norm minimization but am having trouble trying to understand it. I'd also like to understand it at a more intuitive level. [...
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3answers
1k views

Operator norm calculation for simple matrix [closed]

Suppose $$ A = \left( \begin{array}{cc} 1 & 4 \\ 5 & 6 \end{array}\right) $$ How do I calculate $\|A\|_{\text{OP}}$? I know the definition of operator norm, but I am clueless on how to ...
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1answer
222 views

Is the spectral norm submultiplicative?

I wonder if the $2$-norm or spectral norm is also submultiplicative for non-square matrices, i.e., $$\| A B \|_2 \leq \| A \|_2 \cdot \| B \|_2$$ if the number of columns of $A$ coincides with the ...
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6answers
5k views

Why is the Operator Norm so hard to calculate?

I recently took a better look at the operator norm defined on a matrix $\mathbf A \in \Bbb{K}^{n\times n}$ as follows: $$ \|\mathbf A\|_p=\sup\{\|\mathbf Ax\|_p \mid x\in\Bbb{K}^n\land\|x\|=1\} $$ ...
10
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2answers
920 views

What are some usual norms for matrices?

I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix? If so, that would that imply matrices also form some sort of ...
0
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1answer
263 views

Is the spectral norm a Lipschitz function with respect to the spectral norm?

I was wondering if the spectral norm is a Lipschitz function with respect to the spectral norm. How can we prove whether it is or not? In other words, is $$\big| \|X\| - \|Y\| \big| \le L \|X-Y\|$$ ...