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Find the M to meet the Lyapunov equation in (3.59) with What are the eigenvalues of the Lyapunov equation? Is the Lyapunov equation singular? Is the solution unique? Repeat Problem 3.31 for B- Ci- A1 -2 with two different C

3.7 Lyapunov Equation Consider the equation AM +MB C (3.59) where A and B are, respectively, n x n andmx m constant matrices. In order for the equation to be meaningful, the matrices M and C must be of order n × m. The equation is called the Lyapunov equation. The equation can be written as a set of standard linear algebraic equations. To see this we assume n 3 and m 2 and write (3.59) explicitly as m1 12 a21 a22 a23m21 222122 b21 b22 C11 C12 Lc31 C32 Multiplying them out and then equating the corresponding entries on both sides of the equality

Multiplying them out and then equating the corresponding entries on both sides of the equality we obtain an +bn a12 a13 a21a22 +ba23 a31 b12 a32 a33 + bil 0 a11 +b22 a12 a13 a21a22 +b2a23 a32 b12 bi2 d31 a33 + b22 3.8 Some Useful Formulas 71 m11 m21 m31 m12 m22 m32 C11 (3.60) C12 С 32 This is indeed a standard linear algebraic equation. The matrix on the preceding page is a square matrix of order n x m 3 x 2- 6. Let us define A(M):- AM + MB. Then the Lyapunov equation can be written as A(M C. It maps an nm-dimensional linear space into itself. A scalar n is called arn eigenvalue of A if there exists a nonzero M such that Because A can be considered as a square matrix of order nm, it has nm eigenvalues nk, for k-1. 2, ..., nm. It turns out Mk=λί+Hj for i = 1, 2, , n; J=1, 2, , m where λί, 1 = 1 , 2, .. ., n, and μj, j = 1, 2, . . . , m, are, respectively, the eigenvalues of A and B. In other words, the eigenvalues of A are all possible sums of the eigenvalues of A and B We show intuitively why this is the case. Let u be annx 1 righteigenvector of A associated with λί; that is. Au-λίυ. Let v be a 1 x m left eigenvector of B associated with μj; that is, vB - vi, Applving A to the n xm matrix uv vields

ANSWER SHOULD BE NEAT CLEAN AND WELL EXPLAINED.HANDWRITTEN NEAT CLEAN,EACH STEP SHOULD BE EXPLAINED WELL

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