Matrix square matrix

It is well known that the trace of a square matrix (i.e., the sum of its diagonal elements) is unchanged by a similarity transfonnation. If we define the traces... 

Here we have collected the N independent g that correspond to different incoming states for N open chaimels into a matrix g (where the sans serifhoM notation is again used to denote a square matrix). Also we have the matrices... 

Figure 2-16. a) The redundant incidence matrix of ethanal can be compressed by b) omitting the zero values and c) omitting the hydrogen atoms, in the non-square matrix, the atoms are listed in columns and the bonds in rows. 

We can now assign the four carbon p-orbitals, one to each carbon. For simplicity, we will label them with the subscript corresponding to the number of the carbon atom to which the AO belongs. We will use the symbol p to denote AOs and P for MOs. We can now write the Hiickel matrix as a square matrix involving the AOs as shown in Figure 7-20. 

A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

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 ... 

A matrix is symmetric if it is a square matrix with elements such that the elements above and below the diagonal are mirror images = A . 

To compute the inverse of a square matrix it is necessary to first calculate its determinant, A. The determinants of 2 x 2 and 3x3 matrices are calculated as follows ... 

The transpose of a square matrix is, of course, another square matrix. The transpose of a symmetric matrix is itself. One particularly important transpose matrix is the adjoint natris, adJA, which is the transpose matrix of cofactors. For example, the matrix of cofactors ul liie 3x3 matrix... 

I he Fock matrix is a ff x ff square matrix that is symmetric if real basis functions are used. Tile Roothaan-Hall equations (2.149) can be conveniently written as a matrix equation ... 

Each element is designated with a double subscript in general, an element is called fly where j is its horizontal position in the ith row of the matrix. A matrix with m rows and n elements in each row is an m x n matrix. A square matrix with n elements in each row is an n x n matrix. 

If two square matrices of the same size can be multiplied, then a square matrix can be multiplied into itself to obtain A, A, or A". A is the square root of A and the nth root of A". A number has only two square roots, but a matrix has infinitely many square roots. This will be demonstrated in the problems at the end of this chapter. 

The degree of the least equation, k, is called the rank of the matrix A. The degree k is never greater than n for the least equation (although there are other equations satisfied by A for which k > n). If A = n, the size of a square matrix, the inverse A exists. If the matrix is not square or k < n, then A has no inverse. 

Eigenvalues and Eigenveetors of a Square Matrix An eigenveetor of a matrix, M, is a veetor sueh that... 

Given any vector v, we can form a square matrix denoted v(i)Xv(i), whose elements are defined as follows ... 

Program FNCT MAT This program is designed to read in a real square matrix, perform a funetion on it, and return this new array. Possible fun etions, using X as the input matrix, are ... 

The matrix (C) is called the generalized inverse of C. Having estimated the matrix K, one can then estimate the amounts of analytes in an unknown sample. If the number of sensors is equal to the number of analytes, iCis a square matrix. If K exists then... 

The accuracy of C depends on the choice of sensors. As det(R) increases, the agreement between the calculated C and the tme C gets better. To improve accuracy, it is recommended where possible to have more sensors than analytes. In some cases, such as when one sensor s response is a linear combination of the other sensor responses, the number of sensors must be greater than the number of analytes. If the number of sensors is greater than the numbers of analytes, Ris not a square matrix and the sensitivity becomes... 

Inverse of a Matrix A square matrix A is said to have an inverse if there exists a matrix B such that AB = BA = Z, where Z is the identity matrix of order n. 

If A is a square matrix and if principal submatrices of A are all nonsingular, then we may choose P as the identity in the preceding factorization and obtain A = LU. This factorization is unique if L is normahzed (as assumed previously), so that it has unit elements on the main diagonal. 

This represents a system of simultaneous equations equal in number to the number of rows of the square matrix, Each equation consists, on the left, of the sum of the products of the members of a row of the square matrix and the corresponding members of the W-column matrix and, on the right, of the member of that row in the third matrix. With this set of equations solved for Wj, the net flux at any surface Aj is given by... 

In the basic metric matrix implementation of the distance constraint technique [16] one starts by generating a distance bounds matrix. This is an A X y square matrix (N the number of atoms) in which the upper bounds occupy the upper diagonal and the lower bounds are placed in the lower diagonal. The matrix is Ailed by information based on the bond structure, experimental data, or a hypothesis. After smoothing the distance bounds matrix, a new distance matrix is generated by random selection of distances between the bounds. The distance matrix is converted back into a 3D confonnation after the distance matrix has been converted into a metric matrix and diagonalized. A new distance matrix... 

Diagonal matrix This is a square matrix with all elements off the diagonal equal to zero. 

A square matrix is one in which the number of columns is equal to the number of rows. An important type of square matrix which arises quite often in the finite element method is a symmetric matrix. Such matrices possess the property that aij = aji- An example of such a matrix is given below ... 

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) ... 

The determinant of a square matrix of order two is called a determinant of order two and is defined as... 

The cofactor matrix of a square matrix is the matrix of cofactors of each element, i.e.. 

Matrix division is not defined, although if C is a square matrix, C (the inverse of C) can usually be defined so that... 

The determinant of a square matrix C (det C) is defined as the sum of all possible products found by taking one element from each row in order from the top and one element from each column, the sign of each product multiplied by (-ly, where r is the number of times the column index decreases in the product. 

A square matrix is a matrix that contains the same number of columns as rows. 

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. 

If a square matrix has an inverse, the product of the matrix and its inverse equals the unit matrix. The inverse of a matrix A is denoted by A1. 

A square matrix of order n can be considered a vector in a space of n2 dimensions, and if the matrix function j... defines a vector norm in 2-space, it will be called a generalized matrix norm. Thus a generalized matrix norm satisfies... 

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