Operations on Matrices |

**Determinant** > s.a. Berezinian;
characteristic polynomial [eigenvalues].

$ __Cofactor__: The cofactor
of *M _{ij}* is
(–1)

$

det *L*:= (*n*!)^{–1} (–1)^{s} *ε*_{a..
b} *ε*^{c..
d}* L*^{a}_{c} ··· *L*^{b}_{d}
; also, det *M* = ∑_{i}_{ or }_{j}_{ =
1.. }* _{n}* (cofactor

* __Useful formula__: det(I + *tX*) = 1 + tr(*tX*) + *O*(*t*^{2})
= 1 + tr(*tX*) + det(*tX*) (at least for the 2 × 2 case).

* __Derivative__: For a symmetric
matrix, ∂(det *A*)/∂*A*_{ij}
= (det *A*) *A*^{–1}_{ij}.

@ __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.

**Other Operations and ****Related Concepts **> s.a. Commutators.

* __Inverse of a matrix__:
The matrix *M*^{–1} such that *M*^{–1}*M* =
*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} *ε*_{k.. lj} *ε*_{m.. ni} *M _{km}*

* __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 *λ*_{i},
of multiplicity *m*_{i},
rank(*A*–*λ*_{i }I)
= *n*–*m*_{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 *λ* =
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.

* __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}*BA*^{–1}
+ *A*^{–1}*BA*^{–1}*BA*^{–1} – ...

* __Exponentiation__: The
simple exponential e^{A} is defined
in terms of the power series expansion; For a sum, e^{A+B} =
e^{A} e^{B} e^{–[A,B]/2},
provided that *A* and *B* commute with their commutator; > more
generally, see the Zassenhaus Formula.

* __Derivatives__: (*A*^{–1})*'* = –*A*^{–1}*A'A*^{–1},
at least if *A* is symmetric;
∂(det *A*)/∂*A*_{ij} =
(det *A*) (*A*^{–1})_{ji} [notice
the transpose].

* __Resolvent of a matrix__:
The matrix (*λ*I – *M*);
The inverse is
(*λ*I – *M*)^{–1} = *λ*^{–1} + *λ*^{–1}*M* *λ*^{–1} + *λ*^{–1}*M **λ*^{–1}*M* *λ*^{–1} +
... (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].

@ __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].

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