Eigenvalues of a matrix

There is an important theorem in matrix algebra which states that the sum of the eigenvalues of a matrix is equal to the sum of the diagonal matrix elements. Thus... 

Use the theorem of Problem 2.25 to show that the product of the eigenvalues of a matrix equals its determinant. 

Equation (2) is called the characteristic (or secular) equation of A, and its roots are the eigenvalues of A, ak. The problem of finding the eigenvalues of a matrix is intimately connected with its conversion to diagonal form. For if A were a diagonal matrix, then its characteristic equation would be... 

It is a mathematical theorem that the eigenvalue of a matrix (Equation 76), with the largest absolute value, is smaller than the largest of 2 hij and also smaller than the largest of 2 hij. We will call such a sum... 

For stability at a rest point one wishes to show that the eigenvalues of the linearization lie in the left half of the complex plane. There is a totally general result, the Routh-Hurwitz criterion, that can determine this. It is an algorithm for determining the signs of the real parts of the zeros of a polynomial. Since the eigenvalues of a matrix A are the roots of a polynomial... 

How is this a matrix related to the a matrix defined in equation (6.19) Find the eigenvalues of a matrix. Since all the eigenvalues are distinct, a matrix can be diagonalized (a = PDP ) and equation (6.27) is modified as ... 

Let s recall how to find eigenvalues and eigenvectors. (If your memory needs more refreshing, see any text on linear algebra.) In general, the eigenvalues of a matrix A are given by the characteristic equation det(A - Af) = 0, where 1 is the identity matrix. For a 2 x 2 matrix... 

The inertia of a matrix is a triple of integers (n<,Mo,n>) where n< is the number of negative eigenvalues of a matrix, no is the number of zero eigenvalues and n> is the number of positive eigenvalues. The inertia is defined only for matrices with real eigenvalues. A matrix with zero eigenvalues is called a... 

G( ) has no singularities in the energy range E E E2 and so, within that range, the roots E of the equation G(ii)m = Am will all coincide with the desired ADC(2) eigenvalues. The Sturm sequence method of finding the eigenvalues of a matrix... 

Several workers have recently been developing methods to analyze the local stability of a critical point on the basis of the signs of the elements of A.(17 5-40) been shown that all the eigenvalues of a matrix have... 

After eliminating c, find the two roots of the resulting equation and show that they are the same as those given in Eq. (1.96). This technique, which we shall use numerous times in the book for finding the lowest eigenvalue of a matrix, is basically the secular determinant approach without determinants. Thus one can use it to find the lowest eigenvalue of certain N x N matrices without having to evaluate anN x N determinant. 

For future reference we now reformulate the above theory in a way which might appear unfamiliar at first glance but on closer inspection will turn out to be a generalization of the procedure we have used many times to find the lowest eigenvalue of a matrix. We are now interested in finding the sum of the N lowest eigenvalues. The matrix eigenvalue problem in Eq. (5.89) is equivalent to four equations, two of which are... 

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