The_Cambridge_Handbook_of_Physics_Formulas

24 Mathematics
Matrix algebra a
⎛
⎞
a 11 a 12 ··· a 1n
Matrix definition a 21 a 22 ··· a 2n
A = ⎜
⎝
.
. ···
⎟
. ⎠
a m1 a m2 ··· a mn
(2.64)
A
a ij
m by n matrix
matrix elements
Matrix addition C = A+B if c ij = a ij +b ij (2.65)
Matrix
multiplication
C = AB if c ij = a ik b kj (2.66)
(AB)C = A(BC) (2.67)
A(B+C)=AB+AC (2.68)
ã ij = a ji (2.69)
Transpose matrix b ã ij transpose matrix
(AB...N)=Ñ...˜BÃ ˜
(2.70) (sometimes a T ij ,ora′ ij )
Adjoint matrix A † = Ã∗ (2.71)
(definition 1) c (AB...N) † = N † ...B † A † (2.72)
∗ complex conjugate (of
each component)
† adjoint (or Hermitian
conjugate)
Hermitian matrix d H † = H (2.73) H Hermitian (or
self-adjoint) matrix
examples:
⎛
⎞
⎛
⎞
a 11 a 12 a 13
b 11 b 12 b 13
⎜
⎟
⎜
⎟
A = ⎝a 21 a 22 a 23 ⎠ B = ⎝b 21 b 22 b 23 ⎠
a 31 a 32 a 33 b 31 b 32 b 33
⎛
⎞
⎛
⎞
a 11 a 21 a 31
a 11 +b 11 a 12 +b 12 a 13 +b 13
⎜
⎟
⎜
⎟
à = ⎝a 12 a 22 a 32 ⎠ A+B = ⎝a 21 +b 21 a 22 +b 22 a 23 +b 23 ⎠
a 13 a 23 a 33 a 31 +b 31 a 32 +b 32 a 33 +b 33
⎛
⎞
a 11 b 11 +a 12 b 21 +a 13 b 31 a 11 b 12 +a 12 b 22 +a 13 b 32 a 11 b 13 +a 12 b 23 +a 13 b 33
⎜
⎟
AB = ⎝a 21 b 11 +a 22 b 21 +a 23 b 31 a 21 b 12 +a 22 b 22 +a 23 b 32 a 21 b 13 +a 22 b 23 +a 23 b 33 ⎠
a 31 b 11 +a 32 b 21 +a 33 b 31 a 31 b 12 +a 32 b 22 +a 33 b 32 a 31 b 13 +a 32 b 23 +a 33 b 33
a Terms are implicitly summed over repeated suffices; hence a ik b kj equals ∑ k a ikb kj .
b See also Equation (2.85).
c Or “Hermitian conjugate matrix.” The term “adjoint” is used in quantum physics for the transpose conjugate of
a matrix and in linear algebra for the transpose matrix of its cofactors. These definitions are not compatible, but
both are widely used [cf. Equation (2.80)].
d Hermitian matrices must also be square (see next table).