Adjoint

Let A be an n X n matrix and let B, given in Equation 1.35, denote the matrix of its cofactors. Then the adjoint of A, written as adjA, is the transpose 

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If del A 0, then Equation 1.39b will have a unique solution given by 

Because the value oi D 0, the system will have a unique solution. 

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In this case the flucPiation-dissipation relation, ( A3.2.21T reduces to D = IcTa. It is also clear that GE = (A + S)/Mlct which is not self-adjoint. 

The transpose of a square matrix is, of course, another square matrix. The transpose of a symmetric matrix is itself. One particularly important transpose matrix is the adjoint natris, adJA, which is the transpose matrix of cofactors. For example, the matrix of cofactors ul liie 3x3 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 ... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Operators that do not obey the above identity are not hermitian. For sueh operators, it is useful to introduee the so-ealled adjoint operator as follows. If for the operator R,... 

These two operators are not Hermitian operators (although Jx and Jy are), but they are adjoints of one another ... 

Using the above results for the effeet of J+ aeting on j,m> and the faet that J+ and J. are adjoints of one another, allows us to write ... 

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) ... 

Lagrange multipliers are often referred to as shadow prices, adjoint variables, or dual variables, depending on the context. Suppose the variables are at an optimum point for the problem. Perturb the variables such that only constraint hj changes. We can write... 

Adjoint of a matrix The adjoint of an x matrix is the transpose of the matrix when all elements have been replaeed by their eofaetors. 

So far we have not made any assumptions about the properties of the operator A, but we will now assume that A is either self-adjoint or commutes with its hermitean adjoint operator Ht, so that... 

The components of the operator P are hermitian.2 In general, any differential operator Q has a hermitian adjoint Qf, defined by the integral relation... 

These matrices are formed from Q by the rule discussed in Margenau and Murphy, op. cit., p. 374,11.17. They are adjoints in the sense of Chapter 10 of that work. 

An operator is called the adjoint of if they are related as follows ... 

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 ... 

Upon taking the adjoint of Eq. (9-147) we deduce that ( x) anni-hflates the vacuum when operating on it from the right ... 

Since e is the eigenvalue of a hermitian operator, it is real hence, upon taking the hermitian adjoint of Eqs. (9-208) and (9-209) we deduce that... 

Upon taking the hermitian adjoint of this equation, we obtain... 

Since the matrices — yu,yt , "/lT, and y" all obey the same commutation rules as -/ (take the hermitian adjoint, transpose and complex conjugate of Eq. (9-254) ) it follows from theorem G, that there exist nonsingular matrices A, B, C, D, such that... 

The matrix y5 = y°y1y2y3[(y5)2 = +1], anticommutes with all the y s. It therefore has the property that ysyu(y5) 1 — — / . Hence, a possible choice for D is y6. Further properties of the matrices A,B,D can be obtained as follows Consider for example the matrix A. Upon taking the hermitian adjoint of Eq. (9-259) and substituting therein Eq. (9-259) again, we obtain... 

Equation (9-369) allows us to infer that the transformation of the adjoint spinor under Lorentz transformation is given by... 

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