Unitary matrix

From Wikipedia, the free encyclopedia

In mathematics, a unitary matrix is an  complex matrix U satisfying the condition

where I_n is the identity matrix in n dimensions and  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 

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,

so also a unitary matrix U satisfies

for all complex vectors x and y, where  stands now for the standard inner product on .

If  is an n by n matrix then the following are all equivalent conditions:

1.  is unitary
2.  is unitary
3. the columns of  form an orthonormal basis of  with respect to this inner product
4. the rows of  form an orthonormal basis of  with respect to this inner product
5.  is an isometry with respect to the norm from this inner product
6.  is a normal matrix with eigenvalues lying on the unit circle.

http://en.wikipedia.org/wiki/Unitary_matrix

Normal matrix

From Wikipedia, the free encyclopedia

A complex square matrix A is a normal matrix if

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*=A^T; it is normal if A^TA = AA^T.

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.

http://en.wikipedia.org/wiki/Normal_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:

If the conjugate transpose of a matrix A is denoted by , then the Hermitian property can be written concisely as

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.

http://en.wikipedia.org/wiki/Hermitian_matrix

Spectral radius

From Wikipedia, the free encyclopedia

In mathematics, the spectral radius of a matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ(·).

http://en.wikipedia.org/wiki/Spectral_radius

Krylov subspace

From Wikipedia, the free encyclopedia

In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspacespanned by the images of b under the first r powers of A (starting from A⁰ = I), that is,

It is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper on this issue in 1931.^[1]

Modern iterative methods for finding one (or a few) eigenvalues of large sparse matrices or solving large systems of linear equations avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors. Starting with a vector, b, one computes Ab, then one multiplies that vector by A to find A²b and so on. All algorithms that work this way are referred to as Krylov subspace methods; they are among the most successful methods currently available in numerical linear algebra.

Because the vectors tend very quickly to become almost linearly dependent, methods relying on Krylov subspace frequently involve some orthogonalization scheme, such as Lanczos iteration for Hermitian matrices or Arnoldi iteration for more general matrices.

The best known Krylov subspace methods are the Arnoldi, Lanczos, Conjugate gradient, GMRES (generalized minimum residual),BiCGSTAB (biconjugate gradient stabilized), QMR (quasi minimal residual), TFQMR (transpose-free QMR), and MINRES (minimal residual) methods.

References

• Nevanlinna, Olavi (1993). Convergence of iterations for linear equations. Lectures in Mathematics ETH Zürich. Basel: Birkhäuser Verlag. pp. viii+177 pp.. MR1217705. ISBN 3-7643-2865-7.
• Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM. ISBN 0898715342. OCLC 51266114.

1. ^ Mike Botchev (2002). "A.N.Krylov, a short biography".

そろばん

The soroban (算盤, そろばん^?, counting tray) is an abacus developed in Japan. It is derived from the suanpan, imported from China to Japan around 1600.^[1] Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocketelectronic calculators.


http://en.wikipedia.org/wiki/Soroban

Metzler matrix

 a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero)

Metzler matrices appear in stability analysis of time delayed differential equations and positive linear dynamical systems. Their properties can be derived by applying the properties of Nonnegative matrices to matrices of the form M + aI where M is a Metzler matrix.