Matrices  

In General > s.a. operators.
* Idea: Basically, a matrix is a mathematical spreadsheet.
* History: Matrices were introduced to physicists by the 1925 paper on quantum mechanics by M Born and P Jordan.
* Problem: Halmos 1986, Of the \(2^{n^2}\) n × n matrices A of 0s and 1s, how many have det A = 1? det A = 0? What is max{det A}? 10,000 random 10 × 10 matrices were tested and the largest determinant found was 24.
@ References: Marcus 60; Horn & Johnson 85; Cullen 90; Lewis 91 [II]; Joshi 95 [and physics]; Friedland 15.
> Related topics: see characteristic polynomial [eigenvalues]; operations on matrices [determinant, inverse, etc].
> In physics: see Matrix Mechanics; Matrix Models; thermodynamical systems [entropy].

Jordan Normal Form
$ Def: A matrix of block-diagonal form, each block corresponding to one eigenvalue λi,

\[ \def\_#1{_{_{#1}}}
A = \left(\matrix{B\_1 & 0 &\cdots &0 \cr 0 & B\_2 &\cdots &0 \cr
\vdots &\vdots &\ddots &\vdots \cr 0 &0 & \cdots &B\_n}\right)\;, \quad {\rm where}\quad
B\_i = \left(\matrix{\lambda\_i & 1 &0 &\cdots &0 \cr 0 & \lambda\_i & 1 &\cdots &0 \cr
\vdots &\vdots &\vdots &\ddots &\vdots \cr 0 &0 &0 &\cdots &\lambda\_i}\right)\;. \]

* Applications: It is the equivalent of the diagonal form for a non-diagonalizable matrix, the best one can do.

Random Matrices > s.a. Zeta Function.
* History: The theory can be thought of as originated with Hurwitz's1897 introduction of an invariant measure for the matrix groups SO(N) and U(N); It has been studied by statisticians from the 1930s and mathematical physicists from the 1950s.
@ Introductions, reviews: Forrester et al JPA(03) [rev]; Caselle & Magnea PRP(04); Fyodorov mp/04-ln [intro]; Erdős a1004 [survey]; Eynard et al a1510-ln [intro]; Diaconis & Forrester a1512 [origins].
@ General references: Zinn-Justin PRE(99)mp/98 [addition, multiplication]; Vasilchuk & Pastur CMP(00) [addition]; Cicuta & Mehta JPA(00) [determinants]; Borodin & Olshanski CMP(01)mp/00 [∞]; van Moerbeke in(01)m.CO/00; Janik NPB(02) [multicritical ensembles]; Gudowska-Nowak et al NPB(03)mp [infinite products]; Magnea mp/05-ln [and symmetric spaces]; > s.a. characteristic polynomial; Matrix Models.
@ And physical systems: Bertola & Harnad JPA(06) [integrable systems]; Akemann a1603-ln [QCD]; > s.a. quantum chaos.

Other Special Types > s.a. Hessian; Pfaffian; Normal and Subnormal Matrix.
* Bistochastic matrix: A square matrix with positive entries such that rows and columns sum to unity; A unistochastic matrix is a bistochastic matrix whose matrix are the absolute values squared of a unitary matrix.
@ References: Bengtsson qp/04-conf [bistochastic]; Jarlskog JMP(05)mp, JMP(06) [unitary, parametrization].
> Special matrices: see Hilbert and Weingarten Matrix.


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