All Questions
9 questions
5
votes
1
answer
196
views
What is the "natural" or "physical" norm on the Hessian matrix (and other higher derivatives)?
Let $u : \mathbb R^n \rightarrow \mathbb R$ and let $H : \mathbb R^n \rightarrow \mathbb R^{n \times n}$ be its Hessian matrix. What is the "natural" choice of pointwise norm on the Hessian ...
- 641
2
votes
1
answer
87
views
Symmetric and anti-symmetric matrices and maximal eigenvalues
Suppose we start with a symmetric $n \times n$ matrix $A$, the elements of which are either $1$ or $0$. All the diagonal elements of this matrix are set to be $0$. So, $\lambda_{\text{max}}=\sup \...
- 1,755
5
votes
1
answer
319
views
Analytical form for the nuclear norm of an $n \times n$ matrix
I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is:
$$ \Vert A \Vert_* = \sqrt{\operatorname{tr}(...
- 71
0
votes
0
answers
174
views
Lipschitz map on positive definite cone of $n$-by-$n$ matrices
A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
- 91
1
vote
0
answers
139
views
A lower-bound on matrix-function with vector product
I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...
- 5,405
1
vote
1
answer
128
views
Asymptotic behavior of a matrix equation and its eigenvalues
We have a matrix valued function $A:\mathbb{R}_+\to \mathbb{R}^{m\times m}$. It is known that $A(\lambda)$ is a positive definite matrix for all $\lambda\in\mathbb{R}_+$ Denoting $\rho_i(A(\lambda))$ ...
- 698
2
votes
0
answers
186
views
Bounding the condition number of a matrix associated with an even symmetric positive definite function
Define a set $A = \{x_i/x_i\in\mathbb{R}^m, i = 1,2,3..n\}$. Let $f:\mathbb{R}^m\to(0,\infty)$ be an even symmetric positive definite function.
Let $D = [d_{i,j}]$ be an $n\times n$ matrix such that $...
- 698
2
votes
0
answers
87
views
Conditions for $B$ that make $ADB + (ADB)^T$ positive (semi-)definite
I am trying to find conditions under which this dynamical system converges. At the end of the day, we have something like
$$
0 \leq x^TD_0W_0D_1W_1D_2 \dots W_ND_NCx
$$
With $D_i$ matrices that are ...
- 121
7
votes
1
answer
304
views
Argument principle for matrices
Let $f,g$ be entire functions, then the argument principle teaches us that
$$\frac{1}{2\pi i}\int_{\mathbb{C}} g(z) \frac{f'(z)}{f(z)} dz$$
is equal to $g$ evaluated at the zeros of $f.$
Now, let ...
- 167