Matrix units

The exchange matrix, K, is just the rate, k, times the unit matrix. In block fonn, the full matrix for two sites is given in the eigenvalue equation, (B2.4.38). 

I - FH) r on a quadratic surface. Here I is the unit matrix and denotes the nth cycle. The ultimate convergence rate is governed by the magnitude of the largest eigenvalue of the matrix (I - FH). This will be... 

If there is no approximate Hessian available, then the unit matrix is frequently used, i.e., a step is made along the gradient. This is the steepest descent method. The unit matrix is arbitrary and has no invariance properties, and thus the... 

As the D matrix has to be a unit matrix in order to get a continuous, uniquely defined diabatic matrix, the following integral is quantized as ... 

Here, the force F is a linear combination of the components of R it also has a Gaussian distribution and autocorrelation matrix that satisfies the same properties of R t) as shown in eq. (3), with I (the nxn unit matrix) replacing M [71] ... 

We shall often encotmter square matrices, which have the same number of rows and columns. A diagonal matrix is a square matrix in which all the elements are zero except for those on the diagonal. The unit or identity matrix I is a special type of diagonal matrix in which all the non-zero elements are 1 thus the 3x3 unit matrix is ... 

Multiplication of a matrix by its inverse gives the unit matrix ... 

Inserting the unit matrix, in the form into the left-hand side gives ... 

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... 

The unit matrix plays the same role in matrix algebra that 1 plays in ordinar y algebra. Multiplieation of a matr ix by the unit matrix leaves it unehanged ... 

A diagonal matrix has nonzero elements only on the principal diagonal and zeros elsewhere. The unit matrix is a diagonal matrix. Large matrices with small matrices symmetrically lined up along the principal diagonal are sometimes encountered in computational chemistry. 

The unit matrix, I, with an = 1 and Gy = 0 for i plays the same role in matrix algebra that the number 1 plays in ordinary algebra. In ordinary algebra, we can perform an operation on any number, say 5, to reduce it to 1 (divide by 5). If we do the same operation on 1, we obtain the inverse of 5, namely, 1/5. Analogously, in matrix algebra, if we cany out a series of operations on A to reduce it to the unit matrix and cany out the same series of operations on the unit matrix itself, we obtain the inverse of the original matrix A . ... 

Plijcc the orij ini)l eoeffieient matrix next to the unit matrix. Divide row 1 by 2 ami subtraet the result from row 2, This is a linear operation, and it does not ehange the solution set. 

If the functions Oj are orthonormal, then the overlap matrix S reduces to the unit matrix and the above generalized eigenvalue problem reduces to the more familiar form ... 

Program MOCOEFS This program is designed to read in (from the keyboard) the LCAO-MO eoeffieient matrix and write it out to disk. Alternatively, you ean ehoose to have a unit matrix (as your initial guess) put out to disk. With the program 1 imitations deseribed above, MOCOEFS memory usage is 2744 bytes. 

Solving the previous matrix equation for the coefficients C describing the LC AO expansion of the orbitals and orbital energies 8 requires a matrix diagonalization. If the overlap matrix were a unit matrix would simply diagonalize the... 

In a ZDO computation where the overlap matrix is assumed to be the unit matrix, the diagonal elements of the density matrix... 

Unit matrix in mill equations X Particle size or sieve size cm in... 

If all the elements along the principal diagonal of a diagonal matrix are equal, the matrix is called a scalar matrix. One important scalar matrix has all ones on the principal diagonal and is called the Identity or unit matrix ... 

In the unlikely event that none of the basis functions overlap, then S is a unit matrix. We usually require the LCAO orbitals . to be orthonormal... 

This means that the overlap matrix is a unit matrix, in this case... 

---