Matrix, inverse

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

Division of matrices is not defined, but the equivalent operation of multiplieation by an inverse matrix (if it exists) is defined. If a matr ix A is multiplied by its own inverse matrix, A , the unit matrix I is obtained. The unit matrix has Is on its prineipal diagonal (the longest diagonal from upper left to lower right) and Os elsewhere for example, a 3 x 3 unit matrix is... 

This equation expresses the solution to the set of simultaneous equations in that each of the unknown x terms is now given by a new matrix [A] multiplied by the known y terms. The new matrix is called the inverse of matrix [A]. The determination of the terms in the inverse matrix is beyond the scope of this brief introduction. Suffice to say that it may be obtained very quickly on a computer and hence the solution to a set of simultaneous equations is determined quickly using equation [E.4],... 

The principal use of the inverse matrix is in solution of linear equations or the application of transformations. If... 

Splitting the h2 operator into an excitation and de-excitation part, h2 = e, d, allows the propagator to be written as two property vectors times an inverse matrix, often called... 

Since the dimension of the principal propagator matrix may be large, it is impractical to calculate the inverse matrix in eq. (10.115) directly. In practice the propagator is therefore calculated in two steps, by first solving for an intermediate vector X (corresponding to U in eq. (10.50)). 

Now we are ready to formulate the basic idea of the correction algorithm in order to correct the four-indexed operator f1, it is enough to correct the two-indexed operators fc and f 1 in the supermatrix representation (7.100). The real advantage of this proposal is its compatibility with any definite way of f2 and P7 correction [61, 294], The matrix inversion demanded in (7.99) is divided into two stages. In the fi, v subspace it is possible to find the inverse matrix analytically with the help of the Frobenius formula that is well known in matrix algebra [295]. The... 

Partitioning the operator manifold can lead to efficient strategies for finding poles and residues that are based on solutions of one-electron equations with energy-dependent effective operators [16]. In equation 15, only the upper left block of the inverse matrix is relevant. After a few elementary matrix manipulations, a convenient form of the inverse-propagator matrix emerges, where... 

Solving the inverse matrix and assuming that exists yields the partitioned propagators... 

Any non-singular square matrix A possesses an inverse matrix A defined as... 

Thus, the inverse matrix plays the role of division in matrix algebra. Multiplication of equation (1.33) from the left by B and from the right by C yields... 

Let us examine these symbolic transformations with a view toward seeing how they translate into the required arithmetic operations that will provide the answers to the original simultaneous equations. There are two key operations involved. The first is the inversion of the matrix, to provide the inverse matrix. This is an extremely intensive computational task, so much so that it is in general done only on computers, except in the simplest cases for pedagogical purposes, such as we did in our previous chapter. 

The systematic high-school procedure to solve the set of three equations therefore amounts to the procedure of obtaining the inverse matrix A-1. 

There is no matrix version of simple division, as with scalar quantities. Rather, the inverse of a matrix (A-1), which exists only for square matrices, is the closest analog to a divisor. An inverse matrix is defined such that AA"1 = A-1 A = I (all three matrices are n X n). In scalar algebra, the equation a-b = c can be solved for b by simply multiplying both sides of the equation by la. For a matrix equation, the analog of solving... 

The use of Equation (A. 17) for inversion is conceptually simple, but it is not a very efficient method for calculating the inverse matrix. A method based on use of row operations is discussed in Section A.3. For matrices of size larger than 3 X 3, we recommend that you use software such as MATLAB to find A 1. 

The inverse matrix also can be employed in the solution of linear algebraic equations,... 

Row operations can also be used to obtain an inverse matrix. Suppose we augment A with an identity matrix I of the same dimension then multiply the augmented matrix by A-1 ... 

B and C are both noted A 1 which is called the inverse matrix of A. If A has an inverse, it is said to be regular. If B does not exist, A is said to be singular. The demonstration of the following useful properties will be found in standard textbooks... 

Expressions for the variances and covariance of B and A follow from the inverse matrix (M WM) . See the discussion of case-f" from Table V in the Appendix, for explicit formulas. 

Because this X matrix is orthogonal, the elements of any one column multiplied by the corresponding elements of any other column sum to zero. As a result, the product (X X) produces an identity matrix I multiplied by eight (X X) = 87. Thus, the inverse matrix has the form of the identity matrix multiplied by the reciprocal,... 

The verification that this inverse matrix is correct is left as an exercise. 

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