Operations on Matrices| Operations on Matrices |
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
(and s is the number of − signs in the signature of the metric
used to raise indices), then
det L:= (n!)−1 (−1)s εa.. b εc.. 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.
@ General references: Lehmich et al a1209
[convexity of the function C → f(det C) on positive-definite matrices].
@ 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-conf [computation, and quantum field theory];
Kirsten a1005-in [contour-integration methods];
Seiler & Stamatescu JPA(16)-a1512 [fermionic, loop formula];
> s.a. lattice field theory.
> Related topics:
see Cayley-Hamilton Theorem.
Other Operations and Related Concepts > s.a. Commutators.
* Inverse of a matrix:
The matrix M−1 such
that M−1M =
M M−1 = I; If M
is an n × n matrix, it can be calculated using
(M−1)ij = (det M)−1 (cofactor M)ji = (n−1)!−1 (det M)−1 εk.. lj εm.. ni Mkm ··· Mln .
* Diagonalization: If A is
an n × n matrix, with n distinct real/complex eigenvalues,
use GL(n, \(\mathbb R\)/\(\mathbb C\)); If it has degenerate eigenvalues, it can
be diagonalized iff for each λi,
of multiplicity mi,
rank(A − λi I)
= n−mi;
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 λ = 1, but only one eigenvector, (1, 0);
Generalized procedures: The singular-value decomposition and the Autonne-Takagi
factorization; > s.a. Singular Values.
* 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.
* Products: If
A is an n × m matrix and
B is a p × q matrix, their
Kronecker product A ⊗ B is an np
× mq matrix ("tensor product").
* Expansions:
(A+B)−1
= A−1
− A−1
B A−1
+ A−1
B A−1
B A−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 (λI − M); The inverse is
(λI − M)−1
= λ−1
+ λ−1M
λ−1
+ λ−1M
λ−1M
λ−1
+ ... (which converges for λ 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].
@ Diagonalization: Banchi & Vaia JMP(13)-a1207 [quasi-uniform tridiagonal matrices];
Haber a2009 [three procedures];
Bischer et al JPA-a2012 [simultaneous block diagonalization];
> s.a. eigenvectors and eigenvalues.
@ 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 JPA(06)mp/05 [of su(4) matrices];
Fujii & Oike FEJME-mp/06 [formula];
Childs & Wiebe JMP(13)-a1211 [exponentials of commutators, product-formula approximations].
@ Related topics: Fleischhack a0804,
Friedland a0804,
Fleischhack & Friedland a0811 [Hurwitz product traces, BMV conjecture];
Steeb & Hardy 16 [matrix calculus, problems];
Eldar & Mehraban a1711 [approximating the permanent of a random matrix];
Kramer et al a1802 [new product].
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