Adjoint matrices operator

A density matrix or density operator is defined as a self-adjoint matrix (operator) having positive eigenvalues and trace 1. This implies, of course, certain restrictions on the entries d, ...,d in Eq. (14). Here the term operator has essentially the same meaning as matrix, but is appropriate also in the case of infinite-dimensional vector spaces. 

Let Ao = A - -A2, where Ai and A2 are adjoint or triangular (with a triangular matrix) operators, so that... 

There are several types of matrix operations that are used in the MCSCF method. The transpose of a matrix A is denoted A and is defined by (A )ij = Xji. The identity (AB) = B A is sometimes useful where AB implies the usual definition of the product of matrices. A vector, specifically a column vector unless otherwise noted, is a special case of a matrix. A matrix-vector product, as in Eq. (5), is a special case of a matrix product. The conjugate of a matrix is written A and is defined by (A )jj = (A,j). The adjoint, written as A is defined by A = (A ) . The inverse of a square matrix, written as A , satisfies the relation A(A = 1 where = du is called the identity or unit matrix. The inverse of a matrix product satisfies the relation (AB) =B" A" . A particular type of matrix is a diagonal matrix D, where D,y = y, and is sometimes written D = diag(dj, d2> ) or as D = diag(d). The unit matrix is an example of a diagonal matrix. 

According to the basic principles of quantum mechanics, every physical observable Y within the Dirac theory is described by a self-adjoint (4 x 4)-matrix operator, which can always conveniently be written as... 

The combination of transposition and complex conjugation is called the adjoint operation, indicated by a dagger. A Hermitian matrix is thus self-adjoint. An eigenfunction of this matrix, operating in a function space, may be expressed as a linear combination... 

In order to determine the adjoint operators it is necessary to examine in detail the operator terms of Oi and O2 of Eq. (13.78) these are all multiplicative, and thus self-adjoint except for V Z>iV and V Dj , which may be treated exactly as in Sec. 13.4. The result is that all the individual operators which make up the matrix operators Oi and 0 may be regarded as self-adjoint if we require that DiV(f> y and... 

In reactor perturbation theory, the neutron importance 0+ is the adjoint flux obtained by interchanging rows and columns in the neutron-flux matrix operator and solving. The resulting solution is orthogonal to the flux. 

The matrix of L is the transposed conjugate4 of the matrix of L. It is a useful exercise to show that the analog of Eq. (8-19) is true for the adjoint operator ... 

Difference equations with a symmetric matrix are typical in numerical solution of boundary-value problems associated with self-adjoint differential equations of second order. In what follows we will show that the condition Bi = is necessary and sufficient for the operator [yj] be self-adjoint. As can readily be observed, any difference equation of the form... 

The matrix of a self-adjoint operator in any orthonormal basis is a symmetric matrix. 

As far as a non-self-adjoint operator B is concerned, the workable procedure g reduces to inversion of a lower triangle matrix. ... 

It is clear from the definition that the adjoint operator Ta is independent of the choice of basis, but is also illustrative to show that the expression (2.25) is invariant under the transformation X = X.a. In order to understand the expression (2.25) somewhat better, one should observe that the reciprocal basis Xr has its own metric matrix ... 

If one starts the iterative SCF-procedure from a self-adjoint operator p, the properties (3.50) and (3.49) are going to be invariant under the iterations and are going to characterize the final solution. Since Teff is self-adjoint, the classical canonical matrix t is always on diagonal form. We note that, in this case,... 

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