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Jacobi Method For Complex Hermitian Matrices
In mathematics, the Jacobi method for complex Hermitian matrices is a generalization of the Jacobi iteration method. The Jacobi iteration method is also explained in "Introduction to Linear Algebra" by . Derivation The complex unitary rotation matrices ''R''''pq'' can be used for Jacobi iteration of complex Hermitian matrices in order to find a numerical estimation of their eigenvectors and eigenvalues simultaneously. Similar to the Givens rotation matrices, ''R''''pq'' are defined as: : \begin (R_)_ & = \delta_ & \qquad m,n \ne p,q, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^ \end Each rotation matrix, ''R''''pq'', will modify only the ''p''th and ''q''th rows or columns of a matrix ''M'' if it is applied from left or right, respectively: : \begin (R_ M)_ & = \begin M_ & m \ne p,q \\ pt\frac (M_ e^ - M_ e^) & m = p \\ pt\frac (M_ e^ + M_ e^) & m = q \end \\ pt(MR_^\dagger)_ & = \begin M_ & n \ne p,q \\ \frac (M_ e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrices
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate only, and no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Eigenvalue Algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is named after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but only became widely used in the 1950s with the advent of computers. This algorithm is inherently a dense matrix algorithm: it draws little or no advantage from being applied to a sparse matrix, and it will destroy sparseness by creating fill-in. Similarly, it will not preserve structures such as being banded of the matrix on which it operates. Description Let S be a symmetric matrix, and G=G(i,j,\theta) be a Givens rotation matrix. Then: :S'=G^\top S G \, is symmetric and similar to S. Furthermore, S^\prime has entries: :\begin S'_ &= c^2\, S_ - 2\, s c \,S_ + s^2\, S_ \\ S'_ &= s^2 \,S_ + 2 s c\, S_ + c^2 \, S_ \\ S'_ &= S'_ = (c^2 - s^2 ) \, S_ + s c \, (S_ - S_ ) \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Matrices
In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if U^* U = UU^* = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (), so the equation above is written U^\dagger U = UU^\dagger = I. A complex matrix is special unitary if it is unitary and its matrix determinant equals . For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes. Properties For any unitary matrix of finite size, the following hold: * Given two complex vectors and , multiplication by preserves their inner product; that is, . * is normal (U^* U = UU^*). * is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and matrix multiplication, multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the coordinates of the endpoint of a vector with the length ''r'' and the angle \phi with respect to the -axis, so that x = r \cos \phi and y = r \sin \phi, then the above equations become the List of trigonometric identities#Angle sum and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Givens Rotation
In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory. As action on matrices A Givens rotation acting on a matrix from the left is a row operation, moving data between rows but always within the same column. Unlike the elementary operation of row-addition, a Givens rotation changes both of the rows addressed by it. To understand how it is a rotation, one may denote the elements of one target row by x_1 through x_n and the elements of the other target row by y_1 through y_n: \begin \vdots & \vdots & \ddots & \vdots \\ x_1 & x_2 & \dots & x_n \\ \vdots & \vdots & \ddots & \vdots \\ y_1 & y_2 & \dots & y_n \\ \vdots & \vdots & \ddots & \vdots \end Then the effect of a Givens rotation is to rotate each subvector (x_k,y_k) by the same angle. As with row- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
