By Alexander S. Poznyak (Auth.)
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Additional info for Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques
F an-l), nt- an [ Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 50 Proof. Indeed, )~ 0 -1 )~ 0 -1 9 0 0 0 9 an an-1 9 9 9 PCa ()~) O - d e t _ )~ 9 -1 )~ = ()~ + a l ) det . 9 0 -1 9 ~, + a~ 0 9 9 -1 0 9 . . 9 9 0 )~ 9 -~1 01 i 1 9 )~ 0 - 0 9 ~ O1 det -- a2 0 )~ + .... an-1 0 -1 det 9 0 + an det 0 9 . 01 . . 9 -1 )~ 9 9 0 9 9 -1 0 9 . . 0 9 9 9 . )~ = (/~, q- a l ) ~ n - 1 -I'- a2~, n-2 _qt_. . 11. 15) if ~, - 0. 20) Eigenvalues and eigenvectors 51 and to calculate the coefficient corresponding A in the fight-hand side.
P r o o f Pre-multiplying both equations below 9 Ax(i) -- x(i), . iX(i) derive (Ax(J)) (x(i)*Ax(J)) * - * x (i) -- )~ix(J)*x (i) ( A x ( J ) ) * x (i) -- ~j ( x ( i ) * x ( J ) ) * - J~jx(J)*x(i) Multiplying the second equation by ( - 1 ) and summing both equalities we obtain 0 -- ()L i -- ~ j ) x ( J ) * x (i) V] Since ,ki ~: )~j the result follows. 2. The matrix A -- u vr (u, v ~ It~~• has one eigenvalue ~1 equal to v r u (with the corresponding eigenvector x ~l~ - u) and all other eigenvalues ,kir equal to 0 (with the corresponding eigenvectors x ~ir - w~i#l~lv).
3. 18). Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 36 Proof. 4. Two matrices A and B are said to be e q u i v a l e n t (or belonging to the same equivalent class) if they may be transformed by the elementary operations application to the same canonical matrix form. This is written as A ~ B. 9. 5. A s q u a r e n x n m a t r i x A is said to be s i m p l e if it is equivalent to a diagonal matrix D. These definitions will be used frequently below. 6. For a matrix A e ~mxn the size r (1 < r
Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques by Alexander S. Poznyak (Auth.)