Adjoint matrices matrix

In thi.-. case the adjoint matrix is the same as the matrix of cofactors (as A is a symmetric. njlri.x). The inverse of a matrix is obtained by dividing the elements of the adjoint matrix tlie determinant ... 

Suppose that the problem is to find a B-matris of D such that the variables C, and E each occur in one and only one of the B-vectors. Since the submatris Af of Cconsisting of the first three rows corresponding to the variables C, and E is nonsingular, according to Theorem 6 there exists a B-matrix with the desired property. Let Af be the adjoint matrix of M. Then (eq. 52) ... 

The matrix represented in this chapter by A is usually called the adjoint matrix. It is obtained by constructing the matrix which is composed of all of the cofactors of the elements a,j in A and then taking its transpose. With the basic definition of matrix multiplication (Eq. (29)J and some patience, die reader can verify the relation... 

The adjoint of a matrix is constructed using the cofactors defined earlier. The elements atj of the adjoint matrix A are defined as... 

In other words, the adjoint matrix is the array composed of the transpose of the cofactors. 

An important general concept of matrix algebra is the association of each matrix A with a corresponding adjoint matrix A, defined in (9.14) ... 

Before defining the adjoint matrix we must define the transpose of a matrix. This is a matrix of which the columns are the rows, and vice versa, of the original matrix. Symbolically, the transpose of the matrix [ay] is [a Now, the adjoint matrix of a matrix [ay] is defined as follows ... 

Therefore, the eigenvalues of a Hermitian matrix are real, and the eigenvalues of a skew-Hermitian matrix are pure imaginary. Now consider the eigenvectors x) and x ) belonging to two different eigenvalues a, a of a self-adjoint matrix A. 

A oonjuga complex of matrix A A e txancpoce of matrix A A aer adjoint of matrix A A inverae of matrix A det(. ) as determinant of matrix A Traoe(. ) sum of diagonal efements of matoix A dtf Kronooker delta (equalx 0 if equals 1... 

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. 

A unitary matrix U satisfies li 1 = U1, where U 1 is the inverse and U1 = (U) the adjoint matrix (Magnasco, 2007). A matrix is said Hermitian if U = UL For real elements, unitary and orthogonal matrices coincide, so that we can use either of them indistinctly. 

The notation u = (u u ), v = (v v ) has been used. These simplifications are easy to introduce in any method based upon the utilization of cofactors. This is true whether the cofactors are computed directly, or indirectly as elements of the adjoint matrix of the overlap —provided only that the block diagonal form of is fully exploited. 

These can also be expressed in terms of the adjoint matrix of A as in the biorthogonalization technique of Prosser and Hagstrom . A recently proposed method is a reformulation of this procedure and simply consists of... 

The transpose of the complex conjugate matrix (sometimes termed the adjoint matrix), is written as A and defined such that ... 

In order to write S2E as a quadratic form involving a self-adjoint matrix we introduce the 2N x 2N matrix... 

A = conjugate complex of matrix A A — transpose of matrix A A = = adjoint of matrix A A l inverse of matrix A... 

In direct analogy to the nonrelativistic theory of the spin, the self-adjoint matrix... 

Within the Dirac nomenclature of quantum mechanics the state vector is called ket and denoted as xfr). Such a ket is analogous to a wave function but it need not be specified in an explicit form. The ket may often be represented as a column vector (n x 1 matrix). Then the adjoint matrix—a row vector— represents the bra vector denoted as [x(r, i e. 

The transpose of a square matrix is, of course, another square matrix. The transpose of a synunetric matrix is itself. One particularly important transpose matrix is the adjoint matrix, adjA, which is the transpose matrix of cofactors. For example, the matrix of cofactors of the 3x3 matrix... 

The A matrix is known as the adjugate (or adjoint) matrix of m which may be written adj m. Thus... 

If the matrix elements are complex, the adjoint matrix is defined by complex conjugation of the elements followed by transposition, and is denoted with a superscript f. Hermitian matrices are very common in quantum chemistry, and are defined as being self-adjoint, i.e. A = If all the matrix elements are real, the matrix is called symmetric, i.e. A = A . 

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