Matrices| 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 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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send feedback and suggestions to bombelli at olemiss.edu – modified 11 may 2019