# A solution method for roots of matrix quadratic forms

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## Abstract

The algebraic Riccati equation appears in the areas of optimal control, optimal filtering and estimation, and Lyapunov stability studies in a certain form and in the most general form is the generalized matrix quadratic equation. The solution of the algebraic Riccati equation has been obtained in several ways including numerical integration, difference equations, and by computation of the eigenvectors of a matrix composed of the coefficients of the Riccati equation. A new method of computing the solutions to the algebraic Riccati equation is given which is based on the concept of matrix functions. The sign of a matrix is computed to isolate the signs of the real parts of the eigenvalues. Manipulations on this matrix produce what will be called pseudo-Walsh function matrices, i.e., matrices which obey the recursive relations of Walsh functions. The net effect of this procedure is the changing of the signs of particular eigenvalues. The solutions to the algebraic Riccati equation can then be found by the product of a partitioned block and the inverse of a block of a matrix function which is constructed from the pseudo-Walsh matrices. Examples of the method as applied to the general problem and to the particular engineering problems mentioned above will be given.