Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [matrix-norms]

The tag has no usage guidance.

0
votes
1answer
21 views

On the operator norm

Let $A$ be an $n \times n$ matrix. We want to calculate $$\|A\|^2 = \max_{v \in \mathbb{R}^n, \, |v|^2 =1} |Av|^2$$ a) Show that the attempted problem has a solution. b) If $f(v) = |Av|...
0
votes
2answers
14 views

Relationship of spectral radius to matrix norm

I just read that for a real symmetric matrix, the matrix $(A)$ norm equals the spectral radius $(p)$ to the $n^{th}$ power : $||A||=p^n$. I don't think this is true, is it? If so, where does it come ...
2
votes
0answers
22 views

Compute matrix norm induced by weight l1 vector norm

For a strictly positive collection of weights $\{w_{i}\}$, consider the weighted $l_{1}$ vector norm: $$ ||x||_{W} = \sum_{i}^{N} w_{i}|x_{i}| $$ What is (or more accurately, how would you compute) ...
0
votes
2answers
28 views

Upper bound for induced norm

I would like to obtain a tight upper bound for the following matrix norm: $$ \| I - \frac{x x^T}{\|x\|_2^2} \| $$ where $x$ is a column vector. (Clearly, the second term is a rank-1 normalized ...
0
votes
1answer
60 views

Gradient of trace norm of complex matrix

The problem: Let $S \in \mathbb{C}^{N\times M}$ with $N > M$ and $S^{H}S=\mathbb{I}$, let $\rho$ and $\sigma$ be hermitian matrices of trace $1$ and define the function $D: \mathbb{C}^{N\times M} \...
1
vote
1answer
10 views

Basic Matrix norm question

I am pretty bad at linear algebra, sorry if the question is trivial. $P$ and $N$ are two invertible matrix . If $||P^{-1}N||=C<1$, how can we deduce that $$||(I-P^{-1}N)^{-1}||\leq(1-C)^{-1}$$...
0
votes
0answers
21 views

Minimizing the distance between two matrices using norm.

I want to minimize the equation: $$\min_{M,Z} \|M-Z^TZ\|_p$$ with both $M$ and $Z$ as variable matrices with $M\in \mathbf{S}^+$. I am looking to solve this problem using convex optimization. My ...
0
votes
0answers
13 views

Proving the norm expression using Neumann expansion

I have been trying to solve the problem, but wasn't able to move after applying Neumann expansion. The square matrix F satisfies $||F||<1$. I am trying to show the following. $$ ||(I-F)^{-1}|| \...
2
votes
0answers
58 views

Upper bound on $|\boldsymbol{A}^{-1}\boldsymbol{x}|_1$

Let $\boldsymbol x=[1,0,\cdots,0]^T\in\mathbb R^{n×1}$, $ \boldsymbol A\in\mathbb R^{n×m}$ be invertible, and let $f= |\boldsymbol A^{-1}\boldsymbol x|_1$, where $|(·)|_1$ is the absolute value norm ...
0
votes
0answers
27 views

bounding $\|A^K x\|_2$ more tightly than $\|A\|^k_{op} \|x\|$ when $\rho(A) < 1$

Let $\|\cdot\|$ denote the Euclidean operator norm when applied to matrices, and the $L2$ norm when applied to vectors. Let $\rho(\cdot)$ denote the spectral radius (maximum magnitude of $A$'s ...
1
vote
1answer
19 views

Subordinate matrix norm inequality in research paper where authors replace $||A||_{op}^2$ with $||A^TA||_{op}$

I'm perusing this paper. In page 8, I came across this: My question is about equation 45. Also in this question I don't care about neither $u$ nor $M$, I mentioned them just to give some context. In ...
0
votes
2answers
27 views

Spectral norm and inner product

We know that for a general $N\times n$ matrix $A$, its spectral norm is defined as $$\|A\|=\sup_{x\in S^{n-1}} \|Ax\|_{2},$$ where $x$ is from the unit sphere. My question is that why when $A$ ...
0
votes
1answer
22 views

Submultiplicavity of spectral norm

Is spectral norm (i.e. the maximum singular value of a matrix) submultiplicative? I am absolutely confused. How to express the singular value of the product of matrices in terms of that of the ...
0
votes
0answers
46 views

SDP formulation of dual norm

I know that the dual norm of a matrix can be formulated as a semidefinite program (SDP), i.e., $\|X\|_{2,*}$ is the solution to the following SDP in $Y$: $$\begin{array}{ll} \text{maximize} & Y^T ...
1
vote
2answers
47 views

Prove of $\Vert A \Vert_\infty$ submultiplicativity

How can I prove that $\Vert AB \Vert_\infty\le\Vert A \Vert_\infty.\Vert B \Vert_\infty$ ? What I have already done: $\max_{1\le i \le n}(\sum_{j=1}^n|\sum_{k = 1}^n A_{ik}.B_{kj}|)$ $\le \max_{1\...
3
votes
0answers
35 views

Show that the spectral norm of one matrix is smaller than the other.

Given matrices $$A = \begin{bmatrix} 0 & 0 & 0 & 1/3 \\ 0 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 1 \\ 1/3 & 1/2 & 1 & 0 \end{...
2
votes
0answers
31 views

Show that $ \kappa_2(A) \leq [\kappa_1(A) \kappa_{\infty} (A)]^{1/2}$

Suppose for a matrix $ A \in \mathbb{R}^n$, we have $ \ ||A||_2 \leq ||A^TA||^{1/2}$, where $||.||$ is a norm on $\mathbb{R}^n$ associated to matrix norm on $\mathbb{R}^{n \times n}$ and $||.||_2$ is ...
2
votes
0answers
39 views

Show that $ \ \lambda \leq \|A^TA\|$.

Let $ \lambda $ be the eigenvalue of $A^TA$, $A \in \mathbb{R}^{n \times n}$. Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$ with associated subordinate matrix $\|\cdot\|$ on $\mathbb{R}^{n \times n}$. ...
1
vote
1answer
29 views

Show that $\|Ax\|_2^2 = \lambda \|x\|_2^2$

Let $A \in \mathbb{R}^{n \times n}$, let $\lambda$ be an eigenvalue of $A^TA$ and $x \in \mathbb{R}^n \setminus \{0\}$ be the corresponding eigenvector, then show that $$\|Ax\|_2^2 = \lambda \|x\|_2^2 ...
1
vote
1answer
21 views

Example of a matrix where equality occurs for the relation between infinite and 2 norm

Given $A \in \mathbb R^{m\times n}$, I know that: $$\|A\|_2 \le \sqrt {m} \|A\|_\infty$$ $$\|A\|_\infty \le \sqrt {n} \|A\|_2$$ I am supposed to provide an example of a matrix such that the ...
0
votes
1answer
62 views

When is $\|\boldsymbol A\| \|\boldsymbol A^{-1}\|$ bounded?

According to the sub-multiplicative property of (some) matrix norms, we know that \begin{equation} \| \boldsymbol I \| \leq \|\boldsymbol A\| \|\boldsymbol A^{-1}\| \ , \end{equation} for some ...
1
vote
1answer
25 views

Spectral norm of matrices with complex eigenvalues

Suppose that $M$ is a square, invertible matrix with eigenvalues $\lambda_1, \ldots, \lambda_n$ where the lambda's can possibly be complex. Suppose that $\lambda_{\max}(M)$ is complex valued. How is ...
1
vote
1answer
63 views

What are the definitions of the Matrix norms?

Pardon if this question is a bit uninspiring, but the few resources I've found seem to have different definitions. Can someone please provide a definition of the $L_1$, $L_2$, $L_{\infty}$, and $L_p$ ...
0
votes
1answer
41 views

Minimization of norm distance using SDPs, cone programming, etc.

Building on top of this question Minimization of Frobenius norm using SDPs, I would like to know for what class of norms ($ \Vert \cdot \Vert $ in the following) can we represent the following problem ...
1
vote
0answers
32 views

Derive Lipschitz norm equality.

I am reading the paper "Spectral Normalization for Generative Adversarial Networks". The Lipschitz norm is defined as $$\|f\|_{Lip}=\max \frac{\|f(x)-f(x')\|}{\|x-x'\|}$$ In section 2.1, they claim ...
0
votes
1answer
17 views

Example of application of hilbert-schmidt norm

I'm pretty new to linear algebra, and am learning about norms. I'm having trouble understanding how to apply the Hilbert-Schmidt norm. How would you do it for a simple example like $\begin{bmatrix}0 &...
1
vote
1answer
34 views

Norms of vectors and $1\times n$ matrices

I have a question about vector and induced matrix norms. In our class, we were given an exercise to prove equivalences of norms of vectors and their transpose/conjugate transpose. This is the one ...
0
votes
0answers
21 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 ...
0
votes
1answer
46 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}:\...
0
votes
1answer
58 views

Proving the infinity norm is equal to the maximum value of the vector [closed]

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
0
votes
0answers
36 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\|...
2
votes
0answers
42 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 < \...
1
vote
1answer
31 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
votes
1answer
42 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 ...
1
vote
1answer
40 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
votes
1answer
138 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 $...
3
votes
0answers
41 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}|\...
0
votes
1answer
31 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&...
0
votes
1answer
30 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 ...
0
votes
2answers
20 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
votes
1answer
61 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*...
0
votes
1answer
32 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
votes
1answer
215 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 $...
2
votes
2answers
46 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 $A^tA=\begin{bmatrix}1&2&0\\2&5&2\\0&2&5\end{bmatrix}$ ...
0
votes
1answer
281 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 ...
0
votes
1answer
24 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: $$||\...
0
votes
1answer
32 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 ...
0
votes
2answers
102 views

Frobenius Norm Inequality with SVD [closed]

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 $||\cdot||_F$ ...
1
vote
1answer
79 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 ...
2
votes
2answers
314 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 $\left\lVert \cdot \right\rVert_2$ ...