Principles of Mathematics in Operations Research
50 3 Orthogonality http://www.mathreference.com/top-ms,csi.html http://www.maths.adelaide.edu.au/people/pscott/linear_algebra/la ...
4 Eigen Values and Vectors In this chapter, we will analyze determinant and its properties, definition of eigen values and vecto ...
52 4 Eigen Values and Vectors A = an ai2 ai3 021 <J22 0,23 ^31 «32 «33 Volume=det(A) Fig. 4.1. |det A| = Volume(P). 4.1.2 Pro ...
4.1 Determinants 53 det AT = det A. Example 4.1.14 a c bd = ad — cb a b c d If A is triangular, then det A = Yi7=i a" (det/ = ...
54 4 Eigen Values and Vectors 4.2 Eigen Values and Eigen Vectors Definition 4.2.1 The number X is an eigen value of A, with a co ...
4.3 Diagonal Form of a Matrix 55 4.3 Diagonal Form of a Matrix Proposition 4.3.1 Eigen vectors associated with distinct eigen va ...
56 4 Eigen Values and Vectors Example 4.3.3 From the previous example, A = MO i 1 0 0 I 3 "44 =*Ai = 1, A 3 = Ax = X\x <& ...
4.3 Diagonal Form of a Matrix 57 4.3.2 Repeated Eigen Values with Full Kernels In this case, (recall that d(s) = Yii=i(s ~ A;)ni ...
4 Eigen Values and Vectors 10 r 111 010 , 5- V2 -1 _ = (l,l,0)r I I I • 2 2 2 0 0 1 1 1 1 .2 2 2. , S~^1 AS = "200" 020 004 s ...
4.3 Diagonal Form of a Matrix 59 Proposition 4.3.10 m(s) = ITf=1(s - Xi)mi, then C" = Af[(A - Ai)mi] e • • -®AT[(A - Xk)mk], whe ...
60 4 Eigen Values and Vectors v 2 = (l,2,4)T, B 2 =. Therefore, B = where Ai 10 01 -12 B- 0 2-1 -2 5-2 1 -2 1 =>r'AB 01 -12 0 ...
4.4 Powers of A 4.4.1 Difference equations Theorem 4.4.3 If A can be diagonalized (A — SAS"^1 ), then uk = Aku 0 = (SAS-^iSAS-^1 ...
62 4 Eigen Values and Vectors Ai = 1.0, A 2 = 0.94 =>• V! , V 2 SAS'^1 = |1 -1 1 1.00 Vk Zk 0.95 0.01 0.05 0.99 -ik |1 -1 1 0 ...
4.5 The Complex case 63 and if we generalize to the matrices eM = I + At + {At)^2 (At)^3 2! 3! If we take the derivative of both ...
64 4 Eigen Values and Vectors Im a+ib= r (coxO + i sinO) *- Re a a+ib=a-ib Fig. 4.2. Complex conjugate Definition 4.5.2 A = AH w ...
4.5 Problems 65 Properties: i. A 6 CmXn, 3 unitary M — U 9 U~lAU = T is upper-triangular. The eigen values of A must be shared b ...
66 4 Eigen Values and Vectors coefficient of 0.04 Blue 2 targets/Red 1 firer per unit time and 0.01 Blue 2 targets/Red 2 firer p ...
4.6 Web material 67 http://kr.cs.ait.ac.th/~radok/math/mat/51.htm http://kr.cs.ait.ac.th/~radok/math/mat3/ml32.htm http://kr.cs. ...
68 4 Eigen Values and Vectors http://www.cap-lore.com/HathPhys/Implicit/eigen.html http://www.cs.berkeley.edu/~wkahan/MathHl10/j ...
4.6 Web material 69 http://www.nova.edu/"zhang/OlCommAlgJordanForm.pdf http://www.numbertheory.org/courses/MP274/realjord.pdf ht ...
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