In a similar manner, if A is an eigenvalue of A, then f ( A ) can also be expressed as in .NET framework

Generate QR Code in .NET framework In a similar manner, if A is an eigenvalue of A, then f ( A ) can also be expressed as

In a similar manner, if A is an eigenvalue of A, then f ( A ) can also be expressed as
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f ( A ) = b , + blA +
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Thus, if all eigenvalues of A are distinct, the coefficients bm ( r n = 0,1,.. . , N - 1 ) can be determined by the following N equations:
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If all eigenvalues of A are not distinct, then Eq. ( A . 5 9 ) will not yield N equations. Assume that an eigenvalue A , has multiplicity r and all other eigenvalues are distinct. In this case differentiating both sides of Eq. ( A . 5 8 ) r times with respect to A and setting A = A ; , we obtain r equations corresponding to A i :
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Combining Eqs. ( A . 5 9 ) and (A.601,we can determine all coefficients bm in Eq. ( A . 5 7 ) .
D. Minimal Polynomial of A:
The minimal (or minimum) polynomial m ( h ) of an N x N matrix A is the polynomial of lowest degree having 1 as its leading coefficient such that m ( A ) = 0. Since A satisfies its characteristic equation, the degree of m(A) is not greater than N.
APP. A]
REVIEW O F MATRIX THEORY
EXAMPLE A.14
The characteristic polynomial is
and the minimal polynomial is
m(A) = A - a
since
Notes:
1. Every eigenvalue of A is a zero of m(A). 2. If all the eigenvalues of A are distinct, then c(A) = m(A). 3. C(A)is divisible by m(A). 4. m(A) may be used in the same way as c(A) for the expression of higher powers of A in terms of a limited number of powers of A.
It can be shown that m(A) can be determined by
where d(A) is the greatest common divisor (gcd) of all elements of adj(A1 -A).
EXAMPLE A.15
Then
REVIEW O F MATRIX THEORY
[APP. A
Thus, d(A) = A - 2 and
0 and
m(A) = ( A - I ) ( A - 21) =
E. Spectral Decomposition:
It can be shown that if the minimal polynomial m(A) of an N x N matrix A has the form
then A can be represented by
where Ej ( j = 1,2,. ..,i) are called consrituent matrices and have the following properties:
Any matrix B for which B2 = B is called idempotent. Thus, the constituent matrices Ej are idempotent matrices. The set of eigenvalues of A is called the spectrum of A, and Eq. (A.63) is called the spectral decomposition of A. Using the properties of Eq. (A.641, we have
The constituent matrices E, can be evaluated as follows. The partial-fraction expansion of
=-+kl A-A,
k2 ------ +
A-A,
+-A -ki , A
APP. A]
REVIEW OF MATRIX THEORY
leads to
Then where
Let e,(A) = kjgj(A). Then the constituent matrices E, can be evaluated as
EXAMPLE A.16 Consider the matrix A in Example A.15:
From Example A.15 we have
Then and Then e,(A)= - ( A - 2 ) e,(A)=A-1
E,=e,(A)= -(A-21)=
REVIEW O F MATRIX THEORY
[APP. A
A.8 DIFFERENTIATION AND INTEGRATION OF MATRICES A. Definitions:
The derivative of an m x n matrix A(t) is defined to be the m x n matrix, each element of which is the derivative of the corresponding element of A; that is,
Similarly, the integral of an m x n matrix A(t) is defined to be
EXAMPLE A.17
Then
B. Differentiation of the Product of Two Matrices:
If the matrices A(t) and B(t) can be differentiated with respect to t, then
Appendix B
Properties of Linear Time-Invariant Systems and Various Transforms
B.l CONTINUOUS-TIME LTI SYSTEMS Unit impulse response: h(t)
Convolution: y ( t ) = x ( t ) * h(t) = Causality: h( t ) = 0, t < 0 Stability:
- D
Ih(t)l dt < m
B 2 THE LAPLACE TRANSFORM .
The Bilateral (or Two-sided) Laplace Transform
Definition:
Properties of the Bilateral Laplace Transform:
Linearity: a , x , ( t ) + a,x,(t) -a,X,(s) + a,X,(s), R ' 3 R , n R , Time shifting: x( t - t,) H e-"oX(s), R' = R X(s - so), R' = R + Re(s,) Shifting in s: e"llx(t) Time scaling: x(at) H -X(S), R' = aR la l Time reversal: x( - t) X( -s), R' = -R Differentiation in t : --sX(s), R' 3 R dt fl(s) - R'=R , Differentiation in s: - tx(t ds I 1 Integration: x(r)dr -X(s), R' > R n {Re(s)> 0)
1 Convolution: x,( t ) * x,( t )
X,(s)X,(s), R'
n R,
PROPERTIES O F LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS
[APP. B
Some taplace Transforms Pairs:
6(t)
1 , all s
e-"' cos w&t)
+a)2+w i
, Re(s) > - R e ( a )
The Unilateral (or One-sided) Laplace Transform
Definition:
x,(s)
/mx(t)e-stdr
0 - = lirn ( 0 - E )
Some Special Properties:
Differentiation in the Time Domain:
d n x ( t) dt"
S " X I ( S ) S " - I X ( O - )- S " - 2 X y 0 - )-
. .. -x(n-') (0-
APP. B]
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