Numerical Matrix Analysis
Conjugates
Let z=a+bi be a scalar with a,b∈Z.
Its complex conjugate z∗ is obtained by negating b
(if b=0 then z is real and z=z∗):
z∗=a−bi.
Now let A be a scalar m×n matrix:
A=[aij]m×n=⎝⎜⎜⎜⎜⎛a11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amn⎠⎟⎟⎟⎟⎞
Its hermitian conjugate (or adjoint) A∗ is the n×m matrix with ijth
entry is the complex conjugate of the jith entry of A
(i.e. of the form aij∗=(aji)∗=aji∗):
A∗=[aij∗]n×m=⎝⎜⎜⎜⎜⎛a11∗a21∗⋮an1∗a12∗a22∗⋮an2∗⋯⋯⋱⋯a1m∗a2m∗⋮anm∗⎠⎟⎟⎟⎟⎞=⎝⎜⎜⎜⎜⎛a11∗a12∗⋮a1n∗a21∗a22∗⋮a2n∗⋯⋯⋱⋯am1∗am2∗⋮amn∗⎠⎟⎟⎟⎟⎞
In the case that A is an all-real matrix, then the adjoint of A is simply the
transpose of A (i.e. A∗=AT).
Furthermore, if A=A∗ then A is hermitian, and by definition must be a square matrix.
(If A all-real hermitian, A=AT, and A is said to be symmetric.)
Inner products
An inner product is a binary operation over a vector space which associates to a pair o