Of considerable interest are linear maps that are "isometric", also known as "distance preserving maps". Such a map is also called an "isometry". Let
denote an arbitrary isometric linear map. Recall from the chapter on orthonormal matrices that any isometric map that maps
to
is linear.
The distance preserving nature of isometries also means that angles are preserved. If
are arbitrary vectors, then the dot product is preserved by isometric transformations:
.
The standard basis vectors for
,
, are all of unit length and are all mutually orthogonal:
If
is the matrix that describes the isometric linear map
, then the columns
are also all of unit length and are all mutually orthogonal:
The "Hermitian Transpose" of a matrix is the transpose with the conjugation of complex numbers applied on top:
The orthonormal properties of the columns of
imply that the inverse of
is simply its Hermitian transpose:
. Any matrix whose inverse is its Hermitian transpose is referred to as being "unitary". The key property of a unitary matrix
is that
be square and that
(note that
is the identity matrix). Unitary matrices denote isometric linear maps.
Given a square
matrix
, analogous to how
is symmetric if
,
is Hermitian if
, meaning that diagonally opposite entries of
are complex conjugates of each other.
For example,
is symmetric but not Hermitian, but
is Hermitian but not symmetric.
Given a square
matrix
with real valued entries, the function
is a quadratic function over the entries of
, referred to as a "quadratic form". All terms in a quadratic form have degree 2. For instance, given the quadratic form
,
can be expressed as:
or as
The coefficient of the term
for
is the sum of the
and
entries. It then becomes sensible to split the coefficient of
between the
and
entries, in essence requiring
to be symmetric:
.
Generalizing to complex numbers, consider the quadratic form
, where
is arbitrary. Requiring that
be Hermitian is similar to the requirement that
be symmetric in the case of real numbers.
always returns a real number if
is Hermitian: