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.

> __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*^{−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}
*B A*^{−1}
+ *A*^{−1}
*B A*^{−1}
*B A*^{−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}
+ ... (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].

@ __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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