Linear Algebra/Unitary and Hermitian matrices

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Unitary Matrices[edit]

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 matrices[edit]

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 forms[edit]

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.