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**Additional resources for Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques**

**Example text**

Eigenvalues and eigenvectors . . . . . . . . . . . . . The Cayley-Hamilton theorem . . . . . . . . . . . . The multiplicities and generalized eigenvectors . . . . . . . . 1. The matrix A E C ~• ~ "-- C ~, written as a "- la~l" an is called a vector a ~ C ~• 1. 2. 3. F o r the set o f vectors x~, X 2 ..... Xk E C n a n d elements or1,012. . . Olk E C the f o l l o w i n g notions m a y be introduced: 1. Linearcombinationso f x l, x2 . . . ql_ O/2X 2 __~ . .

1) + 2(~. - 2) = (~. - 1)(k - 2 ) ( k - 3) which implies )~1 - 1,)~2 - 2,)~3 = 3 One of the solutions of the equation A x (i) , X (2) _ 1 , X (2) _ 1 In this e x a m p l e a n o n s y m m e t r i c (i -- 1, 2, 3) is - - )~i x ( i ) 21 1 m a t r i x has real e i g e n v a l u e s a n d the c o r r e s p o n d i n g e i g e n v e c t o r s . T h e n e x t p r o p o s i t i o n s h o w s w h e n this o c c u r s . 5. If a square n x n matrix A is Hermitian or real symmetric, that is, if or A- Av then obligatory all eigenvalues )~j are real.

X~} " {X - - O/lX 1 -~-O/2X 2 . q l _ . . _ql_Olk X k 9 O[i E C , i -- 1 . . . 3) 41 Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 42 3. Some vectors xl, X2 . . . X k E C n are said to be linearly dependent over C if there existogl,Ot2 . . . Ogk E C n ~ 1 7 6 O/lX 1 "31- ~ 2 X 2 --[- . . (~-~[~ >0) OgkXk : 0 --[- Otherwise, they are said to be linearly independent. 4. I f S is a subspace o f C ~, then a set o f vectors xl, X2 . . . X k E C ~ is called a basis f o r sif 9 Xl, x2 .