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Results tagged with matrices
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user 9652
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
2
votes
$\|x\|_0$ approximation for very large matrices
A numerical method and several studies are given in
Exact and Approximate Sparse Solutions of Underdetermined Linear Equations, SIAM J. Sci. Comput., 31(1), 23–44. Sadegh Jokar and Marc E. Pfetsch …
- 12.7k
1
vote
Relation between the QR decomposition of matrix A and the QR of the augmented version of A
I think the relation is not that simple and here is why:
Since $A_1 = Q_1R_1$ we have
$$
A_2 = \begin{bmatrix}A_1\\ I\end{bmatrix} = \begin{bmatrix}Q_1 & 0\\0 & I\end{bmatrix}\begin{bmatrix}R_1\\I\en …
- 12.7k
3
votes
Accepted
Probability for a random positive-semidefinite matrix to not be positive-definite?
For example, for random $N\times n$ matrices ($N>n$) with iid Gaussian entries, there is Theorem 2.6 there which says that for the smallest singular value $s_\text{min}(A)$ it holds that
$$\sqrt{N} - \ … There is also a probability for a lower bound on the smallest singular value: For square (subgaussian) matrices, Theorem 3.2 says that for some $C>0$, $0<c<1$ and any $\epsilon>0$ it holds that
$$\mathbb …
- 12.7k
5
votes
How to generate constant row and column sum matrices?
I don't know if this is what you want, but here are two differnet ways:
In the case $m=n$: Generate a bunch of random permutation matrices $A_i$ and take a random linear combination $\sum_i \alpha_i A_i …
- 12.7k
31
votes
1
answer
4k
views
Determinants of binary matrices
I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. … My question is: what is the distribution of determinants of all $n\times n$ $\{0,1\}$-matrices? …
- 12.7k
5
votes
Accepted
A problem about matrix inverse and regularization methods
There is a lot of research about this. I can recommend the books
Engl, Heinz Werner, Martin Hanke, and Andreas Neubauer. Regularization of inverse problems. Vol. 375. Springer Science & Business Medi …
- 12.7k
20
votes
1
answer
25k
views
When does the $4\times 4$ 'false Sarrus rule' compute the determinant correctly?
If you did this you most probably have seen a student solving this problem by applying the "Sarrus rule for 4 by 4 matrices". … Discussing this with a colleague today, we asked ourselves the following question:
How does the set of 4 by 4 matrices for which this "False Sarrus Rule" gives the correct determinant looks like? …
- 12.7k
4
votes
Sparse approximation of the inverse of a sparse matrix
$\newcommand{\Ag}{\mathcal{A}_\gamma}$
Not a full answer, put probably a fruitful pointer:
In the discretization of infinite dimensional problems one faces (bi-)infinite matrices. … There, the Jaffard algebra $\Ag$ of bi-infinte matrices (for some $\gamma>1$) is
$$
\Ag = \{A \ : \ |a_{k,l}|\lesssim (1+ |k-l|)^\gamma\}.
$$
It can be shown that the Jaffard algebra is inverse closed, …
- 12.7k
6
votes
Nearest matrix orthogonally similar to a given matrix
Not a full answer, but some pointers on how to get a numerical method: If $\|\cdot\|_2$ denotes the spectral norm, then, by unitary invariance, your problem is equal to
$$
\min_{T\in O(n)}\|AT-TB\|_2. …
- 12.7k