Questions tagged [matrix-norms]

This tag is for questions regarding the Matrix Norm, a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

Filter by
Sorted by
Tagged with
-3
votes
0answers
16 views

How do I merge two Frobenius norm of matrix?

Consider a soft thresholding peoblem: $$\mathop{\arg \min}\limits_B \left\| B \right\|_1 + \frac{1}{2\mu } \left( \left\|Y - AB - Z - \Lambda \right\|_F^2 + \left\| L - B - \Gamma \right\|_F^2 \right) ...
0
votes
0answers
16 views

Bounding the norm of the inverse matrix

Very strangely, I do struggle with answering the following, seemingly elementary, question. For a square matrix $A$, is there an upper bound for the norm of its inverse? In terms of e.g. the norm of ...
0
votes
0answers
15 views

Is the canonical isomorphism $\mathbb R^m\otimes\mathbb R^n\to\mathbb R^{m\times n}$ isometric with respect to the Frobenius norm?

Let $p\in\mathbb N$ and $n_1,\ldots,n_p\in\mathbb N$. Assume the tensor product space $\bigotimes_{i=1}^p\mathbb R^{n_i}$ is equipped with the unique inner product $\langle\;\cdot\;,\;\cdot\;\rangle_{\...
0
votes
1answer
37 views

For non-negative definite symmetric matrices, $\mathrm{tr}(AB)\le \mathrm{tr}(A)\mathrm{tr}(B)$

$\DeclareMathOperator{\tr}{\mathrm{tr}}$ Is the inequality in title true for non-negative definite matrices?? I could neither prove this result, nor provide a counter example. Context I was trying ...
2
votes
1answer
34 views

How should I calculate $||\underline{u}-\underline{w}||_{2}$?

I'm trying to calculate $||\underline{u}-\underline{w}||_{2}$ where: $$ u=\begin{bmatrix}1 & 3\\ 2 & 2\\ 3 & 1 \end{bmatrix},\,\,\, w=\begin{bmatrix}3 & 1\\ 2 & 2\\ 1 & 3 \end{...
2
votes
2answers
36 views

The norm $\|S-Q\|_F$ where $Q$ is orthogonal is minimised by $Q=I$

Problem: Suppose that $S$ is symmetric and semi-positive-definite. Let $\|\cdot \|_F$ be the Frobenius norm. Show that $$\|S-I \|_F \leq \|S-Q\|_F$$ for all orthogonal matrices $Q$, where $I$ is ...
0
votes
0answers
24 views

Show that infinitely many vector norms induce the same matrix norm

Consider the vector norm $\|\vec{x}\|=\frac{1}{n}\sum_{j=1}^n|x_j|$, for $\vec{x}\in\mathbb{R}^n$. I have shown that the matrix norm induced by this vector norm is $\|A\|=\frac{1}{n}max_j\big(\sum_{i=...
1
vote
1answer
54 views

What are matrix norms for? [closed]

Can someone explain to me with SIMPLE WORDS what is the utility of matrix norms. Why do we define p-norms for p >= 1? Why is there a 1-norm and infinity-norm and all sorts of other p-norms, isn't the ...
0
votes
1answer
19 views

Spectral norm of block-matrix inequalities

Let $A,B \in \mathbb{R}^{m \times n}$ and let $\|A\|_2=\sqrt{\lambda_{max}(A^\mathsf{T}A)}$ denote the spectral norm of a matrix, where $\lambda_{max}$ is the maximum eigenvalue. Clearly, we have $\|...
0
votes
0answers
21 views

2-norm of a matrix product

I have some problems bounding the 2-norm of a matrix product. Suppose that $\Lambda\in \mathbb{R}^{m\times m}$ is a diagonal matrix with diagonal entries $\lambda_1 \geq \lambda_2 \geq \cdots \geq \...
1
vote
1answer
29 views

Norm of inverse of all one matrix plus a PSD matrix

Consider the operator norm of the following $n \times n$ matrix: $$ \|(I + 11^* + X^*X)^{-1} 11^* (I + 11^* + X^*X)^{-1}\|, $$ where X is a $n \times n$ matrix. Is it bounded by $\frac{1}{n}$? I ...
0
votes
0answers
38 views

Prove the following inequalities, condition number.

I am trying to prove the following inequalities: If ${A + \delta A} $ is invertible, prove 1) $$ \frac{|||(A+\delta A)^{-1}-A^{-1}|||}{|||(A+\delta A)^{-1}|||} ={cond}(A) \frac{|||\delta A|||}{\|...
2
votes
1answer
37 views

Is it possible to have a function that is in $L^{\infty}(T)$ but not in $L^1(T)$?

I know that there are functions that exist in the $L^1$ norm but not in the $L^{\infty}$. I'm having trouble coming up with the converse. Could someone provide me an example?
0
votes
0answers
14 views

Matrix norm properties - inequality between maximum absolute value and L2 norm

I'm studying matrix computations book. I wonder how to prove following: ${max}_{i,j}{|A_{ij}|} <= ||A||_2$
0
votes
1answer
30 views

Find orthogonal projection matrix $P$ of rank $r$ which maximizes $\mathbb E_x\left[\frac{\|Px\|}{\|x\|}\right]$

Let $x$ be a random vector in on the unit sphere in $\mathbb R^p$ with mean $\mu$ and covariance matrix $\Sigma$. Let $r \in \{1,2,\ldots,p\}$. Question. Find an orthogonal projection matrix $P \in ...
5
votes
0answers
199 views

Low-Rank approximation w.r.t. different norms

I am having quite some trouble trying to prove some things about low rank-approximation, my problem is of the form: Given a matrix $A \in \mathbb{R}^{m\times n}$, $k < rk(A),$ find: $$ X = \...
0
votes
1answer
34 views

Frobenius norm product with two inequalities

I have the Frobenius norm of two products, $\lVert AB\rVert_F$ and $\lVert AC\rVert_F$. $A$, $B$, and $C$ are matrices, the dimensions do not matter as long as they are compatible and $B$ and $C$ have ...
1
vote
1answer
52 views

Spectral norm - trace inequality

I am wondering whether the following is true under which assumptions on A and B? $\operatorname{trace}(AB)\leqslant\|A\| \operatorname{trace}(B)$ The matrix norm is the spectral norm here. Maybe ...
0
votes
1answer
18 views

Bounds for Matrix Inner Product based on singular values

Lewis(1995) " The convex analysis of unitarily invariant matrix functions " states the result by von neumann that $\langle X,Y \rangle \leq \langle \sigma_X,\sigma_Y \rangle$. Does anyone know any ...
0
votes
1answer
36 views

How to interpret the norm of a matrix?

What's geometric interpretation of the Euclidean (Frobenius) norm of the matrix? I know that vector norm is a vector's lenght but what is the matrix norm geometrically?
0
votes
0answers
29 views

Logarithmic derivative of matrix function

in my research I ended up with a term of the following form: $$ C(x)^\prime:C^{-1}(x)^\prime $$ In my case the matrix function $C(x)\in\mathbb{R}^{3\times 3}$ and is always s.p.d. so we can rewrite ...
0
votes
1answer
19 views

What sort of matrix norms bound traces of products?

Suppose I have some linear operators $X_1, \dots, X_n$ on $\mathbb{C}^r$ (i.e. $r \times r$ matrices) and some other operators $Y_1^\epsilon, \dots, Y_n^\epsilon$ which are deformations of the $X_i$, ...
1
vote
1answer
25 views

Decompose nuclear norm

Let $A$ and $B$ be two square matrices such that $A^\top B = 0$ and $B^\top A = 0$. How can we show that $$ \|A+B\|_{nuc} = \|A\|_{nuc} + \|B\|_{nuc}$$
2
votes
1answer
67 views

Approximating integral $p$-adic matrices of order 2

Let $A \in GL_n(\mathbb{Z}_p)$. We consider the maximum norm $\| \cdot \|$ on $M_{n \times n}(\mathbb{Q}_p)$, which coincides with the operator norm with respect to the maximum norm on $\mathbb{Q}_p^n$...
1
vote
1answer
63 views

Show the operator norm of $A^T A - I_n$ can be bounded by $3\max(\delta, \delta^2)$

Let $A$ be an $m\times n$ matrix and $\delta>0.$ If all singular values of $A$ are between $1-\delta$ and $1+\delta$, $$1-\delta\leq s_n(A)\leq s_1(A) \leq 1+\delta,$$ prove $$\Vert A^TA-I_n\Vert \...
0
votes
1answer
75 views

Induced matrix norm formula proof

Suppose that $\Bbb R^m$ and $\Bbb R^n$ are equipped with norms $\|\cdot\|_b$ and $\|\cdot\|_a$ respectively. Show that the induced matrix norm $\|\cdot\|_{a,b}$ can be computed by the formula $$\|A\|...
2
votes
0answers
48 views

Nuclear norm bounded by Frobenius norm

We have matrices $A,B\in\mathbb{R}^{n,m}$. Let $P\in\mathbb{R}^{n,n}$ and $Q\in\mathbb{R}^{m,m}$ be the orthogonal projection matrices in the column space of $A$ and the row space of $A$ (or column ...
3
votes
1answer
59 views

Lowest upper bound on matrix norm

Let $A \in \mathbb{R}^{d \times d}$ be an invertible real matrix and $A'$ the matrix obtained from $A$ by setting all diagonal elements to $0$, namely $$A'_{ij} = \begin{cases} A_{ij} & \text{if } ...
3
votes
1answer
30 views

Bound on the $2$-norm of a matrix with a single entry replaced by $0$

Let $A \in \mathbb{R}^{n \times n}$ be a real matrix and $A'$ any matrix obtained from $A$ by replacing a single entry by $0$. It seems to hold experimentally that $\lVert A' \rVert_2 \leq \lVert A \...
3
votes
1answer
43 views

Estimates for the norm of Hessian

Let $u:\Omega \rightarrow \mathbb{R}$ a twice differential function, with $\Omega$ a subset of $\mathbb{R}^n$. Suppose that we have the following: $$D^2u\geq - \dfrac{(1+K^2)^{1/2}}{\epsilon}I$$ ...
1
vote
2answers
29 views

spectral norm of 2 matrices compared to spectral norm of the difference matrix [closed]

In one paper I read, there is a notion that for matrix A and B of the same size (B is the sparsified version of A through random sparsification), the difference between the spectral norm is no greater ...
0
votes
0answers
32 views

Lp norm of a matrix

I need a small clarification regarding the $L_2$ norm of a matrix. On wikipedia, the $L_{p,q}$ norm, for $p=2, q=1$, is defined as: $$||A||_{2,1}=\sum_{j=1}^n\left(\sum_{i=1}^m|a_{i,j}|^2\right)^{\...
1
vote
2answers
44 views

Let $A$ be non singular, prove that $\frac{1}{\|A^{-1}\|} = \min_{\|x\| = 1} \|Ax\|$

Let $A$ be non singular, prove that $\frac{1}{\|A^{-1}\|} = \min_{\|x\| = 1} \|Ax\|$ I know that $\|A\| = \sup_{\|x\| = 1} \|Ax\|$ but I have no idea how to proceed from here.
0
votes
3answers
76 views

Derivative d/dx of ||A*sigmoid(x)||^2,

Here ||A||^2 is the norm function which computes the sum of squares of all elements, x is a column vector.I have tried a lot of ways of computing it but none give me the correct answer. A is a matrix, ...
0
votes
0answers
33 views

The norm of sum of powers of Markov matrix multiply powers of its transpose

I want to show the $L_1$ norm of a matrix $S$ is bounded by something, which is defined as follows. I have tested on some matrices, but I cannot prove it. Let $A$ be a Markov matrix, i.e., $A_{i,j}...
0
votes
0answers
37 views

Equality for matrix norms [duplicate]

Let $A \in \mathbb{C}^{n,n}$ invertible and we use a norm $\|\cdot\|$. Then it holds that $$\|A^{-1}\|= \max_{||x||=1} \|A^{-1}x\|=\max_{\|x\|=1}\frac{1}{\|Ax\|}. $$ I can see, why that holds for ...
1
vote
1answer
35 views

Matrix norm exercises

Let $f,g:\mathbb{R}^{n\times n}\to \mathbb{R}$ be given by $$\begin{aligned} f(A) &= \max_{i=1,...,n}i\cdot \sum_{j=1}^{n}|a_{ij}|,\\ g(A) &= n\cdot \max_{i,j=1,...,n}|a_{ij}| \end{aligned}$$...
0
votes
1answer
153 views

Proof of Matrix Norm Inequality (Hadamard product)

Let $◦$ be the entry-wise (Hadamard) product operator, where for two matrices $$A = (a_{ij} )_{1≤i≤n,1≤j≤m}, B = (b_{ij} )_{1≤i≤n,1≤j≤m}∈ R^{n×m}$$ we define $$A ◦ B := (a_{ij} b_{ij} )_{1≤i≤n,1≤j≤m}...
0
votes
0answers
13 views

Potential Property of operator norm of invertible matrices?

While working through questions that asking me to compute the condition number of an invertible matrix A, I noticed a interesting observation regarding ||A|| and ||A^(-1)||. Say we chose an suitable ...
1
vote
1answer
36 views

Understanding matrix norms and spectral radius

I thought I understood matrix norms and spectral radius after reading proof of $||A||\geq \rho(A)$. However, in the lecture notes I'm given the following line, and asked what is wrong with the ...
0
votes
1answer
30 views

Suggest matrix $A$ with a maximal singular value of $5$

Suggest matrix $A \in M_{4x4}$ such that : $\mbox{rank}(A) = 2$ $A$ is not diagonal maximal singular value of $A$ is $5$ I have tried to guess some symmetric matrices without any luck .
0
votes
1answer
68 views

For symmetric $A$, show that $\lVert A \rVert_2 \leq \lVert A \rVert_\infty$

Let $A$ be a symmetric, real matrix. Show that the following inequality holds true: $$ \lVert A \rVert_2 \leq \lVert A \rVert_\infty $$ where: $$\lVert A\rVert_2 = \sqrt{\rho(A^tA)}$$ $$\lVert A\...
1
vote
2answers
37 views

Show, for a real, symmetric matrix, that $\| A\|_2^2 \leq \frac{n-1}{n} \|A\|_F^2$.

I recently had an exam with the following question that I just couldn't get a start on: Suppose $A \in \mathbb{R}^{n \times n}$ is symmetric and such that $\mbox{tr}(A)=0$. Show that $$\| A \|_2^2 \...
3
votes
0answers
238 views

“Almost Normal” Matrix and Gap between Spectral Radius/Norm

Let's denote $$\Vert{A}\Vert := \max_{x\neq0}\frac{x^* Ax}{x^*x}$$ and let $\rho(A)$ denote the largest absolute value of the eigenvalues of matrix $A$. From basic linear algebra, one could ...
1
vote
0answers
19 views

What does a large spectral norm mean?

Suppose I have a data set $X \in \mathbb{R}^{n \times d}$, where $n$ is the number of samples and $d$ is the number of measurements. I find that the largest eigenvalue of $X^TX$ is a very large number....
0
votes
0answers
54 views

Condition number of a diagonal matrix with norm not induced by an inner product

I am trying to prove the following: For any diagonal matrix $D = \mbox{diag}(d_i)$, we have $$\mbox{cond}(D)=\frac{\max|d_i|}{\min|d_i|}$$ The matrix norm is a norm induced by a vector norm, i.e., ...
0
votes
2answers
39 views

If $A$ is non-singular then there exist $B$ singular close to it

The exercise says: Show that if $A$ is non-singular, then there exist a singular matrix in a neighborhood with radio $\|A^{-1}\|^{-1}$ and center in $A$. Can you help me with some hint for a way ...
2
votes
2answers
61 views

What if $\|A^2\|_\infty=\|A\|^2_\infty$?

I have this exercise: For the matrix norm $$\|A\|_\infty := \max_{1\leq i\leq n}{\sum_{j=1}^n{|a_{ij}|}}$$ show or refute $$\|AB\|_\infty=\|A\|_\infty\|B\|_\infty$$ What happens in the special case ...
3
votes
1answer
72 views

If $||AB-I||=\epsilon<1$ then $||A^{-1}-B||<||B||(\frac{\epsilon}{1-\epsilon})$

I'm stucked in this problem: Prove that if $||AB-I||=\epsilon<1$ then $||A^{-1}-B||<||B||(\frac{\epsilon}{1-\epsilon})$ My process: 1) We have that $||AB-I||=||-(AB-I)||=||I-AB||$. 2) ...
4
votes
0answers
62 views

Inequality for the matrix infinity norm

Consider the matrix $\ell_{\infty} \to \ell_{\infty}$ operator norm for some matrix $A \in \mathbb{R}^{m \times n}$, given by $$ \| A \|_{\infty} := \sup_{x: \| x \|_{\infty} = 1} \| A x \|_{\infty} :=...

1
2 3 4 5