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 *λ*_{i},

\[ \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].**
**>

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