Characteristic equation of matrix

Remark f. Notice that for a linear system the coefficients ao,, ai j,..., in equation (48) represent the coefficients of the characteristic equation of matrix S. For the nonlinear case, these coefficients do not represent a generalization of the Cayley-Hamilton theorem hence the assumption is necessary for the existence of the solution of the NRRP. 

Equation (8.82) is called the characteristic equation of matrix A. When the nth-order determinant in (8.82) is expanded, it gives a polynomial in A (called the characteristic polynomial) whose highest power is A". The characteristic polynomial has n roots for A (some of which may be equal to each other and some of which may be imaginary), so a square matrix of order n has n eigenvalues. 

This polynomial, which is called the characteristic equation of matrix A, has n roots, which are the eigenvalues of A. These roots may be real distinct, real repeated, or complex, depending on matrix A (see Table 2.4). A nonsingular real symmetric matrix of order n has n real nonzero eigenvalues and n linearly independent eigenvectors. The eigenvectors of areal symmetric matrix are orthogonal to each other. The coefficients a, of the characteristic polynomial are functions of the matrix elements and must be delermined before the polynomial can be used. 

The eigenvalues A of the image structure tensor can be used to detect lines, corners or constant grey value regions. The characteristic equation of matrix M is... 

Equation (9-8) is called the characteristic equation of matrix A. The roots to Equation (9-8) are called the eigenvalue spectrum of matrix A. For every eigenvalue A,-, there exists a corresponding eigenvector c,-. Therefore, if there are n eigenvalues, then there is not one eigenvalue equation, but n eigenvalue equations... 

Thus, if the characteristic equation of the invariance matrix of the p order composition of (j) is... 

The second equation is called the characteristic equation of the matrix A. Once A is known, the vector ut is computed as the unit vector solution of the linear system. It is left to the reader to show that the eigenvalues of a 2 x 2 matrix A can be found as a solution of the equation... 

We will show in Sec. 15.3 that the eigenvalues of the 4 matrix are the roots of the characteristic equation of the system. Thus the eigenvalues tell us whether the system is stable or unstable, fast or slow, overdamped or underdamped. They are essential for the analysis of dynamic systems. 

The eigenvalues of A can be find by solving the characteristic equation of (1.61). It is much more efficient to look for similarity transformations that will translate A into the diagonal form with the eigenvalues in the diagonal. The Jacobi method involves a sequence of orthonormal similarity transformations, 12,... such that A(<+1 = TTkAkTk. The matrix Tk differs from the identity... 

A matrix is diagonalizable if it is equivalent to a diagonal matrix D. The characteristic equation of A is invariant under a similarity transformation, for... 

This equation is equivalent to the characteristic equation of the matrix (1.25) and therefore provides, when exactly resolved, all the eigenenergies of the total system in spite of the privileged role played by the matter subsystem in our calculations. We notice that an equation in z2 is effectively obtained owing to... 

The Cayley-Hamilton theorem is one of the most powerful theorems of matrix theory. It states A matrix satisfies its own characteristic equation. That is, if the characteristic equation of an m X m matrix [A] is... 

Stability criteria can be obtained from the elements of the monodromy matrix as follows. The eigenvalues are the roots of the characteristic equation of A(T) and consequently... 

More generally, the characteristic equation of a matrix A having the dimension nx nis written as... 

The eigenvalues of the matrix are equal to the roots of the characteristic equation of the system. 

Remember that the inverse of a matrix has the determinant of the matrix in the denominator of each element. Therefore, the denominators of all of the transfer functions in Eq. (12.21) contain Det[/ + ji(j)Gc(. )] Now we know that the characteristic equation of any system is the denominator set equal to zero. Therefore, the closedloop characteristic equation of the multivariable system with feedback controllers is the simple scalar equation... 

This equation is frequently called the characteristic equation of the matrix, H. The coefficients, Ci, C2,. . . , c can be seen to have the form ... 

The stationary state (18.4.12) becomes unstable when the real parts of the eigenvalues of (18.4.13) become positive. The eigenvalue equation or the characteristic equation of a matrix A, whose solutions are the eigenvalues, is... 

Expanding this determinant yields the characteristic equation of the matrix... 

Structure Tensors in Seismic Data Analysis 65 The characteristic equation of the above matrix is the cubic polynomial... 

Similar to the above discussions, non-trivial solutions are only possible when the matrix — co M + Ap is singular. Setting the determinant of this matrix to zero gives the characteristic equation of the undamped system ... 

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