By Alexander S. Poznyak (Auth.)
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Additional info for Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques
A s (s < n) be the distinct eigenvalues of a matrix A and xl, Xs denote corresponding eigenvectors. Suppose that there exist numbers o~, o~2. . . ors such that Proof x2 . . . 16) i=1 Show that this is impossible. To prove that c~i - 0 (i - 1. . . 16) on the left by ( A - )~s_~I,• - )~s_2Inxn) " " " ( A - )~llnxn) and noting that ( A - )~ I,• (A - )~s-lln• - - 0, we get " " " (A - ~s-2In• ~-~OliX i ~lln• i=1 = (A - )~s-lln• - )~s-2In• (A - s )~2Inxn) Z(Zii=2 = (A - Zs_lInxn)(A s -- )~s_2Inxn) ' ' ' ( A - )~3Inxn) x ~--~0~/- )~1)()~/- )~2)oe/x/ .
4) Xn} are orthonormal. 4. 7) Eigenvalues and eigenvectors 43 By the definitions above, it is clear that both Ker A and ImA are subspaces in C n and ~ ( A ) , respectively. 1. ql_rank A2 It follows directly from the definitions above. 2. 8). It is not difficult to verify that for any A : $1 c (~n I $4 c__ C p one has rank A + def A - dim(S1) [ 2. for any S C S1 dim A (S) - dim S - dim (S N Ker A) ] 3. f o r a n y A, B : S 1 c C n l > S2 c C m Ker A A Ker B C Ker(A + B) [ 4. If S = S1 then dim A (S) > dim S - def T > d e f A B [ def A B <_ def A + d e f B [ 6.
F - BI~. In• -- A ) ( B n - 1 ) f -1 -+- Bn_2~. n-2 -+-... + BI)~ + Bo) = (~n + al) n-1 + . . + an-1)~ + an)In• Comparing coefficients, we obtain Bn-1 - In• B n - 2 - A Bn_ t -- a l In • Bo -- A B I = an-11n• - A B o -- anln• Advanced Mathematical Tools for Automatic Control Engineers: Volume 1 54 Multiplying the first of these equalities by A n, the second one by A n-l, the jth by A n-j+l and adding them together yields 0 -- A n + al An-1 -+-... 25). 4. If A -1 exists (A is nonsingular), then A -1 = __~ (A n-1 -+-al An-2 -+-...
Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques by Alexander S. Poznyak (Auth.)