Matrix Eigenvalues and Eigenvectors

Matrices were introduced in 1857 by the mathematician and lawyer Arthur Cayley as a shorthand way of dealing with simultaneous linear equations and linear transformations from one set of variables to another. Hie set of linear inhomogeneous equations (8.35) can be written as the matrix equation 

The determinant of a square matrix A is the determinant whose elements are the same as the elements of A. If det A = 0, the matrix A is said to be nonsingidar 

The inverse of a square matrix A of order n is the square matrix whose product with A is the unit matrix of order n. Denoting the inverse by A , we have 

One can prove that A exists if and only if det A 0. (For efficient methods of computing A see Press et ai. Sec. 2.3 Shoup, Sec. 3.3 Problem 8.43. Many spreadsheets have a built-in capability to find the inverse of a matrix.) 

The linear variation method is the most commonly used method to find approximate molecular wave functions, and matrix algebra gives the most computationally efficient method to solve the equations of the linear variation method. If the functions /i, in the linear variation function j) = C / are made to be orthonormal, 

We now find the approximate wave functions corresponding to these Ws. Substitution of Wi = 0.1250018/r /m/ into the set of equations (8.68a) corresponding to (8.71) gives (after division by h ) 

To solve the homogeneous equations (8.73), we follow the procedure given near the end of Section 8.4. We discard the second equation of (8.73), transfer the 4 term to the right side, and solve for the coefficient ratio we get 

Using W2, W3, and W4 in turn in (8.55), we find the following normalized linear variation functions (Prob. 8.38), where X = xjl  

The set of linear inhomogeneous equations (8.35) can be written as the matrix equation 

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The principal topics in linear algebra involve systems of linear equations, matrices, vec tor spaces, hnear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. 

An extension of this result is that a whole family of matrices, each member related to the original M matrix, can be formed by combining die eigenvalues and eigenvectors as follows ... 

Square matrices and tensors can be characterized by their eigenvalues and eigenvectors. If M is an n x n square matrix (or tensor), there is a set of n special vectors, e, each with its own special scalar multiplier A for which matrix multiplication of a vector is equivalent to scalar multiplication of a vector ... 

This is not so for the matrix eigenvalue problem the eigenvalues (and eigenvectors) of real matrices n generally can only be found in the complex plane C (and in C"). The... 

Equations (24) and (27) show that the eigenvalues and eigenvectors of real orthogonal matrices occur in CC pairs. Furthermore, if x) is real, eqs. (24) and (27) imply that a = a, that is that the eigenvalues that correspond to real eigenvectors of real, orthogonal matrices are also real. In fact, because a2 1,... 

Hereafter, the complex matrices are denoted by a wave sign.) Its eigenvalues and eigenvectors are determined through a (complex) diagonalization as... 

Method for the Calculation of Eigenvalues and Eigenvectors of Large Real-Symmetric Matrices Root Homing Procedure . 

As a result of the Lanczos algorithm wc obtain matrices Q/v and T/v. Note that, in general cases, wc can nui the Lanczos algorithm until / = L— 1, where L < N. In this case we obtain L x L matrices Q/ and T, which can still be effectively used to evaluate the eigenvalues and eigenvectors of the original matrix A (Golub and Van Loan, 1996). 

Once the B (or G) and F matrix elements are known for a given symmetry species, the solution of the secular equation leads to the desired eigenvalues and eigenvectors. It should again be noted that the dimensionality of the B or F matrices is reduced to that of an asymmetric unit in the unit cell. 

Informally, an ADT is a data type together with an associated collection of operations on the type. These operations are the only operations permitted for manipulation of objects of the type. Variables of the type record the state of the ADT, while the operations define the behaviour of the ADT. For instance, a matrix is an example of a type which is commonly used in quantum chemical computations. Associated operations could include constructing a matrix adding two matrices producing the inverse of a matrix and diagonalising a matrix to find its eigenvalues and eigenvectors. 

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