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Results tagged with matrices
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user 78061
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.
8
votes
Hankel determinant of incomplete gamma functions
For example, this was studied by Kurt Johansson in the paper Shape Fluctuations and Random Matrices, Commun. Math. Phys. 209, 437 – 476 (2000), and also by Iain M. …
- 2,552
7
votes
Accepted
Are random circulant matrices almost orthonormal?
The diagonal elements of $P=\frac{1}{N}MM^T$, like
$$P_{11}=\frac{1}{N}\sum_{i=1}^NX_i^2,$$
satisfy $ \langle P_{11}\rangle=1$ and $ \langle P_{11}^2\rangle=1+2/N$ (variance decreases like $N^{-1}$).
…
- 2,552
7
votes
Accepted
Expected value of the largest singular value of a random matrix with entries in $N (0,1)$
If $A$ is a Gaussian random matrix as you describe, then the ensemble of matrices given by $A^TA$ is known as the Wishart ensemble, or the Laguerre ensemble. … You can start collecting references about largest eigenvalues by looking here:
Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices, by Pierpaolo Vivo, Satya N. …
- 2,552
3
votes
A question on determinant of a matrix polynomial
Variable $t$ is a polynomial in $|\lambda|$. The even powers are real polynomials in $x,y$. The odd powers are real polynomials in $x,y$ multiplied by $|\lambda|=\sqrt{x^2+y^2}$. The quantity $P^*P$ a …
- 2,552
0
votes
Probability of positive definiteness of a random matrix
(2011)
Index Distribution of Gaussian Random Matrices (2009)
They compute the probability that all eigenvalues of a random matrix are positive. …
- 2,552