Nonsingular matrix equations

Looking at the matrix equation Ax = b, one would be tempted to divide both sides by matrix A to obtain the solution set x = b/A. Unfortunately, division by a matrix is not defined, but for some matrices, including nonsingular coefficient matrices, the inverse of A is defined. 

We first do a demonstration of similarity transform. For a nonsingular matrix A with distinct eigenvalues, we can find a nonsingular (modal) matrix P such that the matrix A can be transformed into a diagonal made up of its eigenvalues. This is one useful technique in decoupling a set of differential equations. 

Proof. Inserting the expression for - given in Lemma 3 into the left-hand side of (34) and multiplying the equation thus obtained by the inverse of the nonsingular matrix H we arrive at the system of matrix equations, whose left-hand sides are the linear combinations of the linearly independent matrices E, Syield system of Eqs. (39). The assertion is proved. 

The system matrix of equation (6.108) contains the two diagonal blocks /3X and j3y and the two offdiagonal blocks Ax and Ay which are both banded tridiagonal. Rather than inverting a tridiagonal block as naively suggested earlier, it is much less costly to multiply the two sides of the block matrix equation (6.108) from the left by the nonsingular block matrix... 

Equation (3.3) can be simplified further by the change of variables q = T p, where T is the transpose of a nonsingular matrix T to be determined shortly. Introducing this change in (3.3) results in... 

A system of n linear equations in n unknowns will have a unique solution, if the determinant of the coefficient matrix is nonsingular, i.e., if A 0. The rows and columns of a nonsingular matrix are linearly independent in the sense that no row (or column) is a linear combination of other rows (or columns). If the coefficient matrix is singular, the equations may have an infinite number of solutions, or no solutions at all, depending on the constant vector. As an illustration, consider the equations ... 

This equation represents the decomposition of a nonsingular matrix A into a unit lower triangular matrix and an upper triangular matrix. Furthermore, this decomposition is unique [2]. Therefore, the matrix operation of Eq. (2.73) when applied to the augmented matrix [A I c] yields the unique solution ... 

The inverse of a matrix can be found by solving the equation A A = 1. Examples of a nonsingular matrix and its inverse are... 

Stewart s argument provides a prescription for constructing a solution of equations (11.61) - (11.63) provided the matrix Is nonsingular for all relevant values of jc, and provided the differential equations (11.64) and (11.65) have solutions consistent with their boundary conditions. It is possible, in principle, to check the nonsingularity of for any... 

Note that the Jacobian matrix dh/dx on the left-hand side of Equation (A.26) is analogous to A in Equation (A.20), and Ax is analogous to x. To compute the correction vector Ax, dh/dx must be nonsingular. However, there is no guarantee even then that Newton s method will converge to an x that satisfies h(x) = 0. 

The C steps whose coefficients are set equal to zero and must correspond to C columns in (16) whose C x C matrix of coefficients is nonsingular. This choice will guarantee that the resulting mechanism is direct. Furthermore, each coefficient which we set equal to zero gives rise to a linear equation in the variables [Pg.289]

Lemma 1. Let a G-invariant ansatz be of the form (22). Then there is q x q matrix H(x) = ( ) nonsingular in 12 satisfying the matrix partial differential equation... 

The determinant of A is nonzero if A is nonsingular, so the solutions to the two detenninantal equations must be the same. B A is the inverse of A 1B, so its characteristic roots must be the reciprocals of those of A" B. There might seem to be a problem here since these two matrices need not be symmetric, so the roots could be complex. But, for the application noted, both A and B are symmetric and positive definite. As such, it can be shown (see Section 16.5.2d) that the solution is the same as that of a third determinantal equation involving a symmetric matrix. 

If there are n0 open channels at energy E, there are n linearly independent degenerate solutions of the Schrodinger equation. Each solution is characterized by a vector of coefficients aips, for i = 0,1, defined by the asymptotic form of the multichannel wave function in Eq. (8.1). The rectangular column matrix a consists of the two n0 x n0 coefficient matrices ao, < i Any nonsingular linear combination of the column vectors of a produces a physically equivalent set of solutions. When multiplied on the right by the inverse of the original matrix a0, the transformed a-matrix takes the canonical form... 

Following a similar procedure to the one employed above, it is easy to verify that we obtain a model that approximates the fast dynamics of the system in Figure 4.2, in the form of Equation (4.20). Also, it can be verified that only 2N + 8 of the 2N + 9 steady-state constraints that correspond to the fast dynamics are independent. After controlling the reactor holdup Mr, the distillate holdup Md, and the reboiler holdup MB with proportional controllers using respectively F, D, and B as manipulated inputs, the matrix Lb (x) is nonsingular, and hence the coordinate change... 

An alternative method to solving Equation 12.3 is to reduce both Rj and R2 to square, nonsingular, nonidentity matrices by projecting each matrix independently onto the space formed jointly by the two matrices. This permits calculation of A and F, via the QZ algorithm [23] and, by extension, relative concentration estimates, Z, and estimates of the true underlying factors in the X- and Y-ways. This is known as the generalized rank annihilation method (GRAM) [24, 25],... 

When written with the help of the Tl matrix as in (19), from (20) the OR parameter and other linear response properties are seen to afford singularities where co = coj, just like in the SOS equation (2). Therefore, at and near resonances the solutions of the TDDFT response equations (and response equations derived for other quantum chemical methods) yield diverging results that cannot be compared directly to experimental data. In reality, the excited states are broadened, which may be incorporated in the formalism by introducing dephasing constants 1 such that o, —> ooj — iT j for the excitation frequencies. This would lead to a nonsingular behavior of (20) near the coj where the real and the imaginary part of the response function varies smoothly, as in the broadened scenario at the top of Fig. 1. 

As has been seen, the operation of forming the derivative of a vector is equivalent to a transformation of this vector into a new vector and that K is a matrix representation of this transformation. As one might expect, the n X n matrix K is not the only matrix that transforms vectors with n elements into their derivatives. Multiplying each side of Eq. (11) of text from the left by an arbitrary n X n matrix P, which has an inverse P (nonsingular), and using the fact that the unit matrix I = PP may be placed at any point in the equation without changing its value, we obtain... 

It must be noticed the cascade interconnection between the algebraic and the differential items. At each time t, the two algebraic equations system admits a unique and robust solution for in any motion in which S (the Jacobian matrix of ( ) is nonsingular for S 0). Feeding the solution into the differential equation, the following differential estimator, driven by the output and the input signals u, is obtained ... 

Equation (18) is easily solved for X if the matrix t WD is nonsingular. If, however, the moments are not sensitive to one or more of the coordinates, or if two or more coordinates are nearly linearly dependent,... 

Equation (2.71) has a unique solution when its homogeneous version has only the zero solution. Since the latter occurs only when r = n, we conclude that, for the case ofm = n, i.e., the number of equations is the same as the number of unknowns, a unique solution occurs regardless of b when A is nonsingular. When m n and r = n, a. unique solution exists for Ax = b as long as the ranks of matrices A and [A b] are the same, where the augmented matrix [A b] is defined by... 

In the class of methods proposed by Broyden, the partial derivatives df/dxj in the jacobian matrix are evaluated only once. In each successive trial the elements of the inverse of the jacobian matrix are corrected by use of computed values of the functions /. Throughout the development which follows, it is supposed that the functions / are real variable functions of real variables and that the functions are continuous and differentiable. If the jacobian matrix B in the Newton-Raphson equation [Eq. (15-3)] is nonsingular, then B 1 exists and... 

The solution is represented by a column vector that is equal to the matrix product A C. In order for a matrix to possess an inverse, it must be nonsingular, which means that its determinant does not vanish. If the matrix is singular, the system of equations cannot be solved because it is either linearly independent or inconsistent. We have already discussed the inversion of a matrix in Chapter 9. The difficulty with carrying out this procedure by hand is that it is probably more work to invert an n by n matrix than to solve the set of equations by other means. However, with access to Mathematica, BASIC, or another computer language that automatically inverts matrices, you can solve such a set of equations very quickly. 

Extending these observations to any number of equations and unknowns, we have the following For any set of n linear equations in n unknowns, if the determinant of the coefficient matrix is zero, then the equations are not all linearly independent and the equations have no unique solution. Any square matrix A having IAI = 0 is said to be singular. Inversely, if the coefficient matrix is nonsingular, so IAI 0, then the equations are linearly independent and a unique solution exists. 

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