By Alexander S. Poznyak (Auth.)
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Additional resources for Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques
P). 28 Advanced Mathematical Toolsfor Automatic Control Engineers: Volume 1 (A | B) -1 -- A -1 @ B -1 ] provided that both A -1 and B -1 exist. 2. x. 2. ) - det [pT (I~x. x. x~ | A) = det (diag (A, A . . . 4 Submatrices, partitioning of matrices and Schur's formulas Given matrix A _ [aij]i,j=l , m , n if a number of complete rows or columns of A are deleted, or if some complete rows or complete columns are deleted, the new matrix that is obtained is called a submatrix of A. A division of matrices into submatrices is referred to as a partition of the matrix.
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 __~ .
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. The second identity n tr A - f~'~i=l Ai may be easily proven using the so-called diagonal form transformation which can be done below. 3. 20). 12. The spectrum o f a unitary matrix U lies on the unit circle.
Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques by Alexander S. Poznyak (Auth.)