Stack Exchange Network

Stack Exchange network consists of 174 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
27 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
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
votes
1answer
54 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{...
2
votes
1answer
49 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
25 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
83 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
44 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
39 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
21 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
19 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
58 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 $||\cdot||_F$ ...
1
vote
1answer
73 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 ...
1
vote
1answer
94 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 $\left\lVert \cdot \right\rVert_2$ ...
0
votes
0answers
19 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.
0
votes
1answer
23 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:...
1
vote
0answers
17 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*...
0
votes
2answers
29 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 \\ \...
0
votes
0answers
19 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 $c$ ...
0
votes
0answers
17 views

Block matrices inequality

I want to find the relation between 1 with 2 and 3. I know the relation between 2 and 3 but I want to know how 2 is smaller than 1, also whats the relation between 3 with 1.
3
votes
0answers
48 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}...
2
votes
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 ...
0
votes
0answers
67 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 ...
2
votes
0answers
39 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\\ \...
0
votes
1answer
65 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 ...
1
vote
2answers
72 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 ...
2
votes
0answers
63 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 ...
0
votes
1answer
64 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. ...
0
votes
1answer
87 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 ...
1
vote
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\...
0
votes
1answer
47 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{||...
0
votes
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'...
9
votes
1answer
290 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 ...
1
vote
1answer
91 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
votes
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 \...
4
votes
1answer
124 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. [...
3
votes
3answers
727 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 ...
-1
votes
1answer
189 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 ...
26
votes
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
votes
2answers
844 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
votes
1answer
244 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 $$\big| \|X\| - \|Y\| \big| \le L \|X-Y\|$$ ...
9
votes
1answer
4k views

Are matrix $p$-norms unitary invariant?

Consider a matrix $X \in \mathbb{R}^{N \times N}$. Let $\| X \|_p$ be its $p$-norm $$\| X\|_p = \left( \sum_{ij} |X_{ij}|^p \right)^{\frac 1p}$$ Is $\|X\|_p$ unitary invariant? That is, given any ...
8
votes
2answers
12k views

Meaning of the spectral norm of a matrix

Is there an intuitive meaning for the spectral norm of a matrix? Why would an algorithm calculate the relative recovery in spectral norm between two images (i.e. one before the algorithm and the other ...
37
votes
3answers
42k views

What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are ...