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 RMS(11)-a1004 [survey];
Eynard et al a1510-ln [intro];
Diaconis & Forrester a1512 [origins];
Livan et al book(18)-a1712.

@ __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 ln(17)-a1603 [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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