The Algebraic Riccati Matrix Equation for Eigendecomposition of Canonical Forms

The algebraic Riccati matrix equation is used for eigendecomposition of special structured matrices. This is achieved by similarity transformation and then using the algebraic Riccati matrix equation to the triangulation of matrices. The process is the decomposition of matrices into small and specially structured submatrices with low dimensions for easy finding of eigenpairs. Here, we show that previous canonical forms I, II, III, and so on are special cases of the presented method. Numerical and structural examples are included to show the efficiency of the present method.

Some Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation

We propose diverse upper bounds for the solution matrix of the continuous algebraic Riccati matrix equation (CARE) by building the equivalent form of the CARE and using some matrix inequalities and linear algebraic techniques. Finally, numerical example is given to demonstrate the effectiveness of the obtained results in this work as compared with some existing results in the literature. These new bounds are less restrictive and provide more efficient results in some cases.