Download e-book for iPad: Advanced Mathematical Tools for Automatic Control Engineers: by Alexander S. Poznyak (Auth.)

By Alexander S. Poznyak (Auth.)

ISBN-10: 0080446744

ISBN-13: 9780080446745

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Extra info for Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques

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Ann] "ann ann = 1 the matrix A becomes the unit (or identity) matrix 0 1 9 0 9 0 9 . O0 . 1 (usually, the subindex in the unit matrix definition is omitted). lxn :--- 0 0 0 2. The matrix A z 9 ~n• [ A r - [aii ]~iiml [ 9 0 " 0 9 0 = is said to be transposed to a matrix A 9 ]~xTM if a22 : ... : Matrices and matrix operations 3. 9) o f A, that is, adjA "-([Aji]~,i=l) "r 4. A square matrix A 9 R ~• is said to be singular or n o n s i n g u l a r according to whether det A is zero or nonzero. 5.

8) we have det A*A = det In xn = 1 det a * a -- (det a*) (det A) = (det A) (det A) - Idet a l - 1 22. 7) a - ' B C (Irxr -}- o a -I B C ) - ' D A - ' Indeed, the simple matrix multiplication implies [A - 1 - A - 1 B C (C + C D A - ' B C ) - ' C D A - ' ] (A + B C D ) = In• - A -~ BC (C + C D A -~ BC) -1 CD + A - 1 B C D - A - 1 B C (C + C D A - ' B C ) -1 [ - C + C + C D A - 1 B C ] D = Inx. - a -1BC (C + C D A - ' B C ) - ' - A -x B C (C + C D A -~ B C ) -~ C D - CD + A - ' B C D A -~ B C D -- Inx,, Analogously, [A-' - A - 1 B C (Irxr + DA -1BC) -1 DA -1] (A q- B C D ) = lnx.

17) x ( i ) * x (j) -- 0 f o r any indices i and j such that J~i r )~j. 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).

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Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques by Alexander S. Poznyak (Auth.)


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