Unitary matrix

A unitary matrix in a linear algebra complex square matrix in which row and column vectors pairs are orthonormal to each other. Thus, the inverse of a unitary matrix is also its adjoint. The set of unitary matrices of fixed size forms with the matrix multiplication as linking the unitary group. By multiplying by a unitary matrix both the Euclidean norm, and the standard scalar product of two vectors is obtained. A real unitary matrix is called orthogonal matrix. Unitary matrices are used in the case of the singular value decomposition of the discrete Fourier transform and in quantum mechanics.

Definition

A complex square matrix is called unitary if the product gives the identity matrix with its adjoint matrix, ie

Applies. If the row vectors of the matrix denoted by, then this condition is equivalent to saying that the standard scalar product of two row vectors

Results, the Kronecker delta is. The row vectors of a unitary matrix thus form an orthonormal basis of the coordinate space. This is also true of the column vectors of a unitary matrix because with the transpose matrix is unitary. In addition, the adjoint of a unitary matrix is unitary, it is therefore

Examples

The matrix

Is unitary, because it is

The matrix

Is unitary, because it is

In general, any orthogonal matrix is unitary, because for matrices with real entries corresponds to the adjoint of the transpose.

Properties

Inverse

A unitary matrix is always invertible and its inverse is equal to its adjoint, ie

The inverse of a matrix is in fact the very one matrix for which

Applies. It is the converse and each matrix whose adjoint is equal to its inverse, is unitary, since it is then

Invariance of norm and scalar product

Is a unitary matrix is multiplied by a vector, the norm of the vector does not change, that is,

Next is the standard scalar product of two vectors is invariant with respect to multiplication with a unitary matrix, ie

Both properties follow it directly from the displacement property of Standardskalarprodukts. Conversely, the picture matrix mapping each linear

Received by the Euclidean norm or the standard scalar, unitary, because it is

Where the -th standard basis vector. Such a linear map is called unitary map accordingly.

Determinant

For the value of the determinant of a unitary matrix

What using the product theorem on determinants

Followed.

Eigenvalues

The eigenvalues ​​of a unitary matrix have also all the complex amount one, ie are of the form

With. Indeed, if an eigenvector associated to, the following applies with respect to a result of the invariance of the Euclidean norm and the absolute homogeneity of a standard

And therefore. A unitary matrix therefore has at most the real eigenvalues ​​.

Diagonalizability

A unitary matrix is normal, that is to say is valid

And therefore diagonalizable. According to the spectral theorem, there is another unitary matrix, so that

Holds, where a diagonal matrix with the eigenvalues ​​of is. The column vectors of are then pairwise orthonormal eigenvectors of. This means that the eigenspaces of a unitary matrix are pairwise orthogonal.

Standardize

The spectral norm of a unitary matrix

For the Frobenius norm applies accordingly with the Frobenius inner product

The product with a unitary matrix receives both the spectral norm, as well as the Frobenius norm of a given matrix, because it is

And

This means that the condition of a matrix with respect to these standards after multiplication by a unitary matrix is preserved.

Preserving the idempotency

Is a unitary matrix and an idempotent, that is valid, then the matrix

Also idempotent, since

Unitary Matrices as a group

The set of regular matrices of fixed size forms with the matrix multiplication as linking a group, the general linear group. As a neutral element here is the identity matrix. The unitary matrices form a subgroup of the general linear group, the unitary group. The product of two unitary matrices is unitary, namely again, because it is

Further, the inverse of a unitary matrix is also unitary, because it is

The unitary matrices with determinant one in turn form a subgroup of the unitary group, the special unitary group. The unitary matrices with determinant minus one form any subgroup of the unitary group, because they lack the neutral element, but merely incidental to class.

Use

Using a singular value decomposition can be any matrix as a product

A unitary matrix, a diagonal matrix and the adjoint of another unitary matrix represent. The diagonal entries of the matrix are then the singular values ​​of.

Unitary matrices are also often used in quantum mechanics in the context of matrix. Examples are:

• The Dirac matrices
• The Pauli matrices
• The S- matrix
• The CKM matrix

Another important application of unitary matrices is the discrete Fourier transform of complex signals.