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 2n^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; Joshi 95 [and
physics].
> In physics: see Matrix
Mechanics, Matrix Models [in physics]; thermodynamical
systems [entropy].
Determinant > s.a. Berezinian; characteristic
polynomial [eigenvalues].
$ Cofactor: The cofactor
of Mij is
(–1)i+j (determinant of
the minor obtained deleting row i and column j from M).
$ Def: If L is a linear map L: V → V, with dim V = n,
then
det L:= (n!)–1 
a..
b c..
d Lac ··· Lbd
; also, det M = 
i or j =
1.. n (cofactor M)ij Mij .
* Useful formula: det(I + tX) = 1 + tr(tX) + O(t2)
= 1 + tr(tX) + det(tX) (at least for the 2 × 2 case).
* Derivative: For a symmetric
matrix, 
(det A)/Aij
= (det
A) A–1ij.
@ Functional determinant: Gursky CMP(97)
[Laplacian and Dirac operator squared]; Elizalde
JHEP(99)ht;
Illies CMP(01)
[regularized products]; Fry IJMPA(02)
[fermion, status]; Kirsten & McKane
AP(03)mp [countour
integration], JPA(04)mp [general
Sturm-Liouville problems]; Dunne JPA(08)-a0711-in
[computation, and quantum field theory]; > s.a. lattice
field theory.
Other Operations and Related Concepts > s.a. Commutators.
* Inverse of a matrix:
The matrix M–1 such that M–1M =
M M–1 = I; Can be calculated
using (M–1)ij =
(det M)–1 (cofactor M)ji.
* Diagonalization: If A is
an N × N matrix,
with N distinct real/complex eigenvalues, use GL(N, R/C);
If it has degenerate eigenvalues,
it can
be diagonalized iff for each 
i,
of multiplicity 
i,
rank(A–i I)
= N–i;
Otherwise one can only reduce to Jordan normal form,
with one Jordan block per eigenvector; Example: A = (1 1 ;
0 1), which has a doubly degenerate eigenvalue =
1, but only one eigenvector, (1, 0).
* Generalization: Any
real symmetric or complex hermitian positive-definite N ×
N matrix is congruent to a diagonal one mod an SO(m, n),
resp SU(m, n), matrix, for any partition N = m + n [@
Simon et al mp/98].
* Decomposition: Every
non-singular matrix can be written as the product of a symmetric one and an
orthogonal one.
* Expansions: (A+B)–1 = A–1 – A–1BA–1
+ A–1BA–1BA–1 – ...
* Exponentiation: The
simple exponential eA is defined
in terms of the power series expansion; For a sum, eA+B =
eA eB e–[A,B]/2,
provided that A and B commute with their commutator; > more
generally, see the Zassenhaus Formula.
* Derivatives: (A–1)' = –A–1A'A–1,
at least if A is symmetric;
(det A)/Aij =
(det A) (A–1)ji [notice
the transpose].
* Resolvent of a matrix:
The matrix (I – M);
The inverse is
(I – M)–1 = –1 + –1M –1 + –1M –1M –1 +
... (converges for sufficiently
large).
* Permanent of a matrix: A
number obtained from an analog of the minor expansion of the determinant, but
with all positive signs; For a unitary matrix, its magnitude is 
1;
> s.a.
knot invariants [application].
@ Inverse: Penrose PCPS(55) [generalized].
@ Factorization: Mostafazadeh mp/02 [symmetric];
Dita JPA(03)
[unitary].
@ Exponentiation: Suzuki PLA(90),
PLA(93)
[of sum]; Federbush mp/99,
LMP(00)mp;
Ramakrishna
& Zhou mp/05 [of
su(4) matrices]; Fujii & Oike mp/06 [formula].
@ Related topics: Fleischhack a0804,
Friedland a0804,
Fleischhack & Friedland a0811 [Hurwitz product traces, BMV conjecture].
Jordan Normal Form
$ Def: A matrix of block-diagonal form, each block corresponding to
one
eigenvalue i,
* Applications: It is the equivalent of the diagonal form for a non-diagonalizable matrix, the best one can do.
Special Types > see Hessian; Pfaffian; Normal and Subnormal
Matrix.
* Random: Studied by
statisticians from the 1930s and mathematical physicists
from the 1950s; > s.a. Zeta Function.
* 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.
@ Random: Guhr et al PRP(98);
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]; Forrester et al JPA(03)
[rev]; Gudowska-Nowak et al NPB(03)mp [infinite
products];
Caselle & Magnea
PRP(04);
Fyodorov mp/04-ln
[intro]; Magnea mp/05-ln
[and symmetric spaces]; Bertola & Harnad JPA(06)
[and integrable systems]; Bleher a0801-in
[Riemann-Hilbert approach]; > s.a. characteristic
polynomial, quantum chaos.
@ Other types: Bengtsson qp/04-in
[bistochastic]; Jarlskog JMP(05)mp,
JMP(06)
[unitary,
parametrization].
> Special matrices:
see
Hilbert and Weingarten
Matrix.
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sep
2009