Matrix principal diagonal

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. 

In eqn (4.1), g and A-t are 3x3 matrices representing the anisotropic Zeeman and nuclear hyperfine interactions. In general, a coordinate system can be found - the g-matrix principal axes - in which g is diagonal. If g and A, are diagonal in the same coordinate system, we say that their principal axes are coincident. 

Spectra of radicals in a dilute single crystal are obtained for various orientations, usually with the field perpendicular to one of the crystal axes. Each spectrum usually can be analyzed as if they were isotropic to obtain an effective g-value and hyperfine coupling constants for that orientation. Since the g- and hyperfine-matrix principal axes are not necessarily the same as the crystal axes, the matrices, written in the crystal axis system, usually will have off-diagonal elements. Thus, for example, if spectra are obtained for various orientations in the crystal vy-plane, the effective g-value is ... 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

It is possible to choose an axis system in which the D-matrix is diagonal - the principal axes. In many cases, these axes will also be the principal axes of the g-matrix and we will so assume in the following. 

S = 29.5803 0 0 0 1.9907 0 0 0 0.2038 Display the S matrix or the singular values matrix. This diagonal matrix contains the variance described by each principal component. Note the squares of the singular values are termed the eigenvalues. 

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

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. 

If the transpose of a matrix is identical in every element to the original matrix (that is, if A = A), then the matrix is called a symmetric matrix. Thus, a symmetric matrix has all elements AtJ equal to all elements Ajt it is symmetric with respect to its principal diagonal from upper left to lower right. A symmetric matrix is necessarily a square matrix, because otherwise its transpose would have different dimensions and could not be identical to it. 

A diagonal matrix is a square matrix that has all its off-diagonal elements zero the (principal) diagonal runs from the upper left to the lower right. 

Note that a symmetric matrix is unchanged by rotation about its principal diagonal. The complex-number analogue of a symmetric matrix is a Hermitian matrix (after the mathematician Charles Hermite) this has atJ = a, e.g. if element (2,3) = a + bi, then element (3,2) = a — bi, the complex conjugate of element (2,3) i = f 1. Since all the matrices we will use are real rather than complex, attention has been focussed on real matrices here. 

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