# Questions tagged [spectral-norm]

The spectral norm of a matrix is its maximum singular value.

199 questions
Filter by
Sorted by
Tagged with
14 views

### What is the second derivative of the spectral norm of a symmetric matrix?

It is well known that the derivative of a matrix $A$'s 2-norm with respect to $A$ is $$\frac{\partial \sigma_{\max}(A)}{\partial A}=u_1v_1^T,$$ where $u_1,v_1$ is the are the first column/row in the ...
24 views

### How to prove the inequality for the spectral radius and spectral norm of an arbitrary matrix?

How to prove $\rho(A)\leq \min_D||D^{-1} A D||_2$, where $D$ is any invertible matrix, $\rho(\cdot)$ is the spectral radius, $\||\cdot\||$ is the spectral norm for the matrix? Are there any other ...
70 views

### 2-norm of transpose proof

I don't understand the proof of ‖x‖2=‖xT‖2. I know you can use the Cauchy-Schwartz inequality and that you can prove it by first showing ‖xT‖2 ⩽ ‖x‖2 and then ‖x‖2 ⩽ ‖xT‖2. I get proving both ...
51 views

### Find the matrix maximizing a summation of bilinear forms

Given $d, n \in \mathbb{N}$ and $x_k, y_k \in \mathbb{R}^d$, for $1 \leq k \leq n$, we want to find $\arg \max_{A: ||A||_2 = 1} \sum_k \langle x_k, A y_k \rangle$, where $||\cdot||_2$ denotes spectral ...
1 vote
64 views

### Operator norm $4$ different definitions how to prove that $\sup$ is $\max$ and $\inf$ is $\min$ for the last two?

From what I have understood, all the $\sup$'s and $\inf$'s in the $4$ different definitions of the operator norm can be taken as $\max$'s and $\min$'s. For a linear map $A\in \mathscr L (V,W)$ between ...
1 vote
75 views

422 views

### What's a sphere in the space of matrices?

Given a normed vector space $V$ over a field $k$ it's sphere is the set of points of unit norm. $$S(V) = \{ v \in V \mid 1 = \lVert v \rVert^2 \}$$ Over the reals the n-spheres are just spheres of ...
1 vote
28 views

### A matrix whose spectral norm is logarithmic in its size?

I'm trying to find a kernel function $f(x,x')$ such that the N by N kernel matrix $A_{i,j} = f(x_i,x_j)$, where $x_i=i/N$, has a spectral norm (largest singular value) of $O(log N)$. Could anyone find ...
1k views

### On the gradient of spectral norm of matrix

Let $\| A \|_2 := \sqrt{\lambda_{\max}(A^TA)}$. As part of the gradient of a regularized loss function (for machine learning), I need the gradient $\nabla_A \| A \|_2^2$, which, using the chain rule, ...
For a random matrix $A\in\mathbb{R}^{n\times m}$ with entries $A_{ij}\sim \mathcal{N}(0,1)$ i.i.d., the spectral norm of it can be bounded by $\|A\| \leq C(\sqrt{m} + \sqrt{n}+t)$ with probability $1-... 1 vote 1 answer 76 views ### Why are we able to define the Frobenius norm as so? I have seen that, in order to prove $$||A||_{2} \leq ||A||_{F}$$ this inequality holds, proofs will use the following definition for a Frobenius norm. $$||A||_{F}^{2} = \sum_{j=1}^{n} ||Ae_{j}||_{2}^{... -1 votes 1 answer 112 views ### Can we deduce this property for 2 norm with respect to sub matrix of Cholesky factorization? [closed] We have A \in \mathbb{R}^{n \times n} which is symmetric and positive-definite. Also, A is a block matrix:$$A = \begin{pmatrix} A_{11} & A_{21}^{\top} \\ A_{21} & A_{22} \\ \end{pmatrix}.$...
How does one proceed in solving this inequality? X is of full column rank \left\|\left(\boldsymbol{X}^{T} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{T} \boldsymbol{v}\right\|_{2} \leq \frac{1}{\...