A matrix with real characteristic roots

The Characteristic Roots of a Matrix*

If A is a square matrix of order n and I is the unit matrix, the equation in X obtained by equating to zero the determinant \A— \l\ is called the characteristic equation of A. The roots of this equation are called the characteristic roots of A. Although it is not possible to make any definite statement as to the nature of the characteristic roots of the general algebraic matrix A, several autho...

On Matrix Polynomials with Real Roots

It is proved that the roots of combinations of matrix polynomials with real roots can be recast as eigenvalues of combinations of real symmetric matrices, under certain hypotheses. The proof is based on recent solution of the Lax conjecture. Several applications and corollaries, in particular concerning hyperbolic matrix polynomials, are presented.

Characteristic Roots and Field of Values of a Matrix

From (1) it follows tha tX = x^4x*. The set of all complex numbers zAz* where zz* — \ is called the field of values [25] of the matrix A. It follows that the characteristic roots of A belong to the field of values of A. Beginning with Bendixson [3] in 1900, many writers have obtained limits for the characteristic roots of a matrix. In many cases these were actually limits for the field of value...

Limits for the Characteristic Roots of a Matrix

Let A be a square matrix of order n with complex numbers as elements. The equation |XJ—A | = 0 is called the characteristic equation of the matrix A, and the roots X;, the characteristic roots of the matrix A. Although it is not possible to make any definite statements regarding the nature of the characteristic roots for the general matrix, several authors have given upper limits to the roots. ...

Real Roots!