Diagonal, principal

To do this, multiply the binomials at the top left and bottom right (the principal diagonal) and then, from this product, subtract the product of the remaining two elements, the off-diagonal elements (42 — 9x). The difference is set equal to zero ... 

The transpose A of a matrix is obtained by reflecting the matrix through its principal diagonal ... 

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. 

A tridiagonal matr ix has nonzero elements only on the pr incipal diagonal and on the diagonals on either side of the pr incipal diagonal. If the diagonals on either side of the principal diagonal are the same, the matrix is a symmetr ic tr idiagonal matr ix. 

This is eontinued n — 1 times until the entire eoeffieient maPix has been eonverted to an upper triangular matrix, that is, a maP ix with only zeros below the principal diagonal. The b veetor is operated on with exactly the same sequenee of operations as the eoeffieient matrix. The last equation at the very bottom of the triangle, aititXit = bn, is one equation in one unknown. It ean be solved for whieh is baek-substituted into the equation above it to obtain x i and so on, until the entire solution set has been generated. 

The principal diagonal of the HMO output matrix is the it electron probability density at atomy, where the summation is over all occupied orbitals. This... 

For a square matrix, the principal or main diagonal goes from the upper left-hand comer to the lower right-hand comer of the matrix. Thus, the principal diagonal has elements Ajj. A symmetric (square) matrix has elements that are symmetric about the principal diagonal, that is... 

A diagonal matrix is a square matrix with zero elements everywhere except on the principal diagonal (that is, all off-diagonal elements are zero) ... 

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 ... 

A diagonal matrix is a square matrix in which all of the dements which do not lie on the diagonal are equal to zero. Note that the diagonal (or, more exactly, the principal diagonal) is comprised of all the elements Ay for which i =j. 

In a square nxn matrix A, the main diagonal or principal diagonal consists of the elements for all i ranging from 1 to n. The latter are called the diagonal elements all other elements are off-diagonal. 

It is not too difficult to develop the multiplication table shown as Table 3 (problem 1). It will be noticed immediately that the table is not symmetric with respect to the principal diagonal Therefore, the group is not Abelian and multiplication is not commutative. 

In general, the gl contain a sum of products of gk with k < n, which can be found by means of repeated reductions of lower order. That part of —glMz which cannot be reduced to an expression in terms of gk with a lower index is called by Caspers the irreducible principal diagonal part and is written... 

The last term Tt is the rest of the development, involving all the orthogonal operators, with the exclusion of M. Caspers assumes that the contribution of the terms which are not principal diagonal may be neglected in the case of large fields. He then finds that for large H and t, Q fulfils the integral equation... 

The time dependence of the principal diagonal part —htMz indicates that this term corresponds to the one which is selected by the condition of dynamic irreducibility in Prigogine and Resibois s formalism. However, it is for the complete expression Tr[MzMz(t)] that one is allowed to retain in the development of Mz(t) only the part proportional to Mz and not for Q(/). 

The matrix (2.13) is a square matrix of order n. The elements axva22,...,ann lie on the principal diagonal of A the sum of these elements is called the trace of the square matrix ... 

A square matrix whose elements all satisfy the relation a0 = ajt is called a symmetric matrix. The elements of a symmetric matrix are symmetric about the principal diagonal. An example is... 

A square matrix all of whose elements are zero, except those on the principal diagonal, is called a diagonal matrix. 

For a square matrix, taking the transpose amounts to reflecting the elements about the principal diagonal. A symmetric matrix is equal to its transpose A = A. The transpose of a row matrix is a column matrix, and vice versa. 

List the SALCs so that all those belonging to a given representation occur together in the list. Use this list to label the rows and columns of the secular determinant. Only the elements of the secular determinant that lie at the intersection of a row and a column belonging to the same irreducible representation can be nonzero, and these nonzero elements will lie in blocks along the principal diagonal. The secular determinant will therefore be factored. 

---