Orthogonal matrix, real

For example, according to Hamermesh (see Ref. [11]), the number of real conditions to uniquely determine an (N x N) (complex) unitary matrix is N2, while the number of real conditions to uniquely fix a (real) orthogonal matrix of same dimensions is not N2/2 but N(N + l)/2. 

The group of all real orthogonal matrices of order 3 and determinant +1 will be denoted by 0(3). Such matrices correspond to pure rotation or proper rotation of the coordinate system. An orthogonal matrix with determinant —1 corresponds to the product of pure rotation and inversion. Such transformations are called improper rotations. The matrix corresponding to inversion is the negative of the unit matrix... 

Since our matrix M is symmetric and has real entries, there always exists an orthogonal matrix A which diagonalizes M. 

The existence of such an orthogonal matrix is guaranteed for a real and symmetric matrix M. 

The orthogonal matrix Q transforms the real symmetric metric matrix M to its diagonal matrix of eigenvalues e ... 

State whether each of the following concepts is applicable to all matrices or to only square matrices (a) real matrix (b) symmetric matrix (c) diagonal matrix (d) null matrix (e) unit matrix (f) Hermitian matrix (g) orthogonal matrix (h) transpose (i) inverse (j) Hermitian conjugate (k) eigenvalues. 

An orthogonal matrix is one whose inverse is equal to its transpose A, =A. A unitary matrix is one whose inverse is equal to its Hermitian conjugate A 1 = A. A real orthogonal matrix is a special case of a unitary matrix. 

The first equation in (2.24) states that column vectors / and j of a unitary matrix are orthonormal the second equation states that the row vectors of a unitary matrix form an orthonormal set. For a real orthogonal matrix, the row vectors are orthonormal, and so are the column vectors. 

Such a matrix is called a real orthogonal matrix. Multiplying U7 with eq. (3.1.1) gives... 

This is a most useful result since we often need to calculate the inverse of a 3 x3 MR of a symmetry operator R. Equation (10) shows that when T(R) is real, I R)-1 is just the transpose of T(R). A matrix with this property is an orthogonal matrix. In configuration space the basis and the components of vectors are real, so that proper and improper rotations which leave all lengths and angles invariant are therefore represented by 3x3 real orthogonal matrices. Proper and improper rotations in configuration space may be distinguished by det T(R),... 

Equation (26) imphes that the real eigenvectors of an orthogonal matrix must correspond to eigenvalues with a2 = 1. For real orthogonal matrices, A = A, ... 

Table A1.2. Real eigenvalues of a real 3x3 orthogonal matrix A with real eigenvectors.

In the case of delocalized basis states tpa(r), the main matrix elements are those with 0 = 7 and f3 = 6, because the wave functions of two different states with the same spin are orthogonal in real space and their contribution is small. It is also true for the systems with localized wave functions tpa(r), when the overlap between two different states is weak. In these cases it is enough to replace the interacting part by the Anderson-Hubbard Hamiltonian, describing only density-density interaction... 

The length of the vector a 2 remains invariant under rotation and it is easy to show that R((j>)RT((f>) = E V, where RT is the transpose of R and E is the unit matrix. Real matrices that satisfy this condition are known as orthogonal matrices. The condition implies that [detil()]2 = 1 or that detfi() = 1. Matrices with determinant equal to —1 correspond to rotations combined with spatial inversion or mirror reflection. For pure rotations detR = 1, for all . 

The matrix TT is symmetric and actually a diagonal matrix since the score vectors are orthogonal. Any real symmetric matrix can be written as the product of the transposed orthonormal eigenvector matrix, V, a diagonal matrix of eigenvalues, G, and V. 

An orthogonal matrix S is a real quadratic matrix for which the transpose S is equal to the inverse S. ... 

If A is a real and symmetric matrix (n x n) there is always an orthogonal matrix S by which A can be transformed into a diagonal matrix in which the diagonal elements are eigenvalues to A... 

Some types of matrices that appear frequently are Hermitian matrices, for which A = A anti-Hermitian matrices, for which A = — A and unitary matrices, for which = U The MCSCF methods diseussed in most detail in this review will involve only operations of real matrices. In this case these matrix types reduce to symmetric matrices, for which A = A , antisymmetric matrices, for which A = — A, and orthogonal matrices, for which U = U. A particular type of orthogonal matrix is called a rotation matrix and satisfies the relation Det(R)= -I-1, where Det(R) is the usual definition of a determi-... 

When the direct product of two irreducible matrix representations of a group is reducible, it can be reduced to a direct sum of irreducible representations by cin equivalent transformation with a constant matrix, i.e. the same matrix for all the matrix representatives of the symmetry operators of the group (2). We shall assume the irreducible representations in unitary form then the constant matrix can be chosen as the real orthogonal matrix whose elements are the coupling coefficients occuring in Eq. (5). The orthogonality properties can be expressed as... 

A systematic way to diagonalize a real symmetric matrix H is as follows. Construct an orthogonal matrix Oj such that the matrix Hj = O HOj has zero in place of the off-diagonal elements H12 and //21 of H. (Because H is symmetric, we have =... 

An example of such a matrix was found in Example 2.3. All rotation matrice.s are orthogonal and have real determinants. Since the determinant of the product of two matrices is equal to the product of the two determinants, and because the detemtinant of a matrix and its transpose are equal, the determinant of a real orthogonal matrix must equal 1. For a unitary matrix, U,... 

The matrix R(9, z) is called a real orthogonal matrix, as are all real matrices that satbfy Eq. 2.26. The transformation of Eq. 2.23 is called an orthogonal transformation. 

Therefore the eigenvector mi, belonging to the real eigenvalue Ai = 1, is identical to the direction r of the searched rotation axis. For the calculation of the angle of rotation u) the rotation matrix R (4) will be transformed with an orthogonal matrix C = (ci,C2,C3) to the form Rw... 

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