Tag: complex

Unitary matrix

Unitary matrix

From Wikipedia, the free encyclopedia
In mathematics, a unitary matrix is an ntimes n complex matrix U satisfying the condition
U^{dagger} U = UU^{dagger} = I_n,
where In is the identity matrix in n dimensions and U^{dagger} is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if and only if it has an inverse which is equal to its conjugate transpose U^{dagger} ,
U^{-1} = U^{dagger} ,;
A unitary matrix in which all entries are real is an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner productof two real vectors,
langle Gx, Gy rangle = langle x, y rangle
so also a unitary matrix U satisfies
langle Ux, Uy rangle = langle x, y rangle
for all complex vectors x and y, where langlecdot,cdotrangle stands now for the standard inner product on mathbb{C}^n.
If U , is an n by n matrix then the following are all equivalent conditions:
  1. U , is unitary
  2. U^{dagger} , is unitary
  3. the columns of U , form an orthonormal basis of mathbb{C}^n with respect to this inner product
  4. the rows of U , form an orthonormal basis of mathbb{C}^n with respect to this inner product
  5. U , is an isometry with respect to the norm from this inner product
  6. U , is a normal matrix with eigenvalues lying on the unit circle.

Normal matrix

Normal matrix

From Wikipedia, the free encyclopedia
A complex square matrix A is a normal matrix if
A^*A=AA^*
where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.
If A is a real matrix, then A*=AT; it is normal if ATA = AAT.
Normality is a convenient test for diagonalizability: every normal matrix can be converted to a diagonal matrix by a unitary transform, and every matrix which can be made diagonal by a unitary transform is also normal, but finding the desired transform requires much more work than simply testing to see whether the matrix is normal.
The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

Hermitian matrix

Hermitian matrix

From Wikipedia, the free encyclopedia
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its ownconjugate transpose – that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:
a_{i,j} = overline{a_{j,i}},.
If the conjugate transpose of a matrix A is denoted by A^dagger, then the Hermitian property can be written concisely as
A = A^dagger,.
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of having eigenvalues always real.