All Questions
10 questions
0
votes
1
answer
91
views
Matrix quantization and effect on singular values
Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for
$$
\|
\sigma_i(A)-\...
- 5,405
0
votes
0
answers
99
views
Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix
Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
3
votes
0
answers
173
views
Can the Jordan decomposition of a matrix be computed in a backwards stable way?
Let $PJP^{-1}$ denote the Jordan decomposition of $M$. The matrix $J$ is a direct sum of Jordan blocks; it is unique up to permutation of the Jordan blocks. The matrix $P$ is not unique.
There are two ...
- 4,943
1
vote
1
answer
323
views
How can I find minimum and maximum eigenvalue of non-positive define matrix [closed]
There is a power iteration method, but it only returns the greatest(in absolute value) eigenvalue of matrix. So when we have negative eigenvalues it'll give wrong results.
Is there any method, which ...
- 11
5
votes
1
answer
404
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?
$$\hat{X}=\...
- 1,495
3
votes
0
answers
244
views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,712
4
votes
3
answers
3k
views
Is this inequality involving the Frobenius norm right?
Let $A$ be a generic (or varying) square, real $ n \times n$ matrix. Let $G$ be a fixed $n \times k$ matrix, $k < n.$ Denote by $||.||_F$ the Frobenius norm.
Is it true that $||AG||_F \geq c(G) ||...
- 1,467
9
votes
1
answer
492
views
When are two binary matrices simultaneously equivalent to their transpose?
For any real square matrix $A$ there is an invertible matrix $P$ such that $A^t = P^{-1}AP$. I have two binary ($0,1$) matrices $A$ and $B$. When does there exist a $P$ such that $A^t = P^{-1}AP$ and $...
- 363
4
votes
0
answers
233
views
Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same
I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, i.e.,...
- 1,517
5
votes
3
answers
693
views
Norm of triangular truncation operator on rank deficient matrices
Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$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