By Alexander Graham
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Additional resources for Kronecker Products and Matrix Calculus: With Applications
Example text
Are scalar variables and we exclude the possibility that any element can be a constant or zero. In general we will also demand that these elements are independent. When this is not the case, for example when the matrix X is symmetric, is considered as a special case. The . a,: . -. ,w. ' (G9 reader will appreciate the necessity for these restrictions when he considers the partial derivatives of (say) a matrix X with respect to one of its elements xr5. Obviously the derivative is undefined if xr,.
1 Lao -ar -a2 -a"- is said to be in `companion' form, it has the associated characteristic equation IA! t + ... 35) (Ch. 3 Some Applications of the Kxonecker Product Example 3. 7 Determine the feedback matrix K so that the two input - two output system 0 1 x= 3 3 2 -3 0 0 0 1 x+ 1 0u 2 0 1 has closed loop eigenvalues (-1, -2, -3). Solution We must first decide on the form of the matrix H. 28)) H - A = BKC and the first row of B is zero, it follows that the first row of H-A must be zero. We must therefore choose H in the companion form.
The Kronecker Product 30 [Ch. 2 The right hand side is a block diagonal matrix, each of the m blocks is the sum I,,,+A+21 A2+... = eA . 17) follows. eA®Im (In®Im)+(A(D Im)+21 Q. ®A)2+... ( 1 n ®In,) + ( A ®1m) + 1(A2 01m) + .. +a.... )trB = tr A tr B . 19) We have seen (Property IX) that if {X;} and {pj} are the eigenvalues of A and B respectively, then {X;pj} are the eigenvalues of the product A ® B. We now show the equivalent and fundamental property for A (D B. XIV If {X;} and tAj) are the eigenvalues of A and B respectively, then (Xi + pf} are the eigenvalues of A O B.