All Questions
12 questions
1
vote
1
answer
331
views
Eigenvalues of a circulant: DFT or Inverse DFT Convention?
Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ...
- 879
1
vote
0
answers
111
views
Solving a block tridiagonal system with diagonal perturbations
Say we have a block tridiagonal matrix, $T \in \mathbb{R}^{NL \times NL}$, with constant off diagonals, $\mathbf{B} \in \mathbb{R}^{L\times L}$, given by
$$
T = \begin{bmatrix} \mathbf{A}_1 & \...
- 11
5
votes
1
answer
401
views
Best orthogonal approximation of rank 1 matrix
Let $X=\lambda_0u_0v_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda_0\in\mathbb{R}$, $u_0,v_0$ are of unit Euclidean norm. What is the solution of the following problem?
$$\hat{X}=\...
- 1,495
0
votes
1
answer
116
views
Matrix equation of the form $C A C^\intercal = D$
Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & ...
- 323
3
votes
0
answers
243
views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,712
8
votes
1
answer
1k
views
Square root of a large sparse symmetric positive definite matrix
I am trying to calculate
$$Y = A^{\frac 12} X$$
where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...
- 83
7
votes
0
answers
197
views
A special eigenvalue problem
For my research I need to solve a generalised eigenvalue problem
$Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{...
- 492
4
votes
3
answers
3k
views
Is this inequality involving the Frobenius norm right?
Let $A$ be a generic (or varying) square, real $ n \times n$ matrix. Let $G$ be a fixed $n \times k$ matrix, $k < n.$ Denote by $||.||_F$ the Frobenius norm.
Is it true that $||AG||_F \geq c(G) ||...
- 1,465
7
votes
1
answer
246
views
Add a multiple of $I$ to a matrix to minimize its operator norm
Given $A\in\mathbb{C}^{n\times n}$, what is $s_* = \arg\min \|A-sI\|$?
Here $\|A\|$ is the operator norm, $\|A\|=\rho(A^*A)^{1/2}$, and $I$ is the identity.
The corresponding problem for the ...
- 20.2k
3
votes
2
answers
216
views
What are interesting heuristics of determining how far given matrix is from a singular one?
The condition number and volume of matrix (defined as absolute value of its determinant) are things which come to mind. Is there more?
I think that over the years numerical folks (who are faced with ...
- 355
5
votes
3
answers
693
views
Norm of triangular truncation operator on rank deficient matrices
Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ A\|\le\frac{\...
- 153
16
votes
2
answers
4k
views
The singular values of the Hilbert matrix
The $n\times n$ Hilbert matrix $H$ is defined as follows
$$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$$
What is known about the singular values $\sigma_1 \geq \cdots \geq \sigma_n$ of $H$?
...
- 3,217