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: VV, 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 Cf(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(Aλi I) = nmi; 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 AB is an np × mq matrix ("tensor product").
* Expansions: (A+B)−1 = A−1A−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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