Linear Algebra/Unitary and Hermitian matrices

Unitary MatricesEdit

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.

Hermitian matricesEdit

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.

Quadratic formsEdit

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:

Theorem

If   is Hermitian, then the quadratic form   always returns a real number.

Proof

 

Since   due to   being Hermitian,   always returns a real number.