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# Eigen values and eigen vectors engineering

3. Dec 2015
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### Eigen values and eigen vectors engineering

1. Linear algebra: matrix Eigen-value Problems Eng. Shubham Kumbhar Part 3
2. Eigenvalue Problems 1. Eigenvalues and eigenvectors 2. Vector spaces 3. Linear transformations 4. Matrix diagonalization
3. The Eigenvalue Problem Consider a nxn matrix A Vector equation: Ax = λx » Seek solutions for x and λ » λ satisfying the equation are the eigenvalues » Eigenvalues can be real and/or imaginary; distinct and/or repeated » x satisfying the equation are the eigenvectors Nomenclature » The set of all eigenvalues is called the spectrum » Absolute value of an eigenvalue: » The largest of the absolute values of the eigenvalues is called the spectral radius 22 baiba jj +=⇒+= λλ
4. Determining Eigenvalues Vector equation » Ax = λx  (A-λΙ)x = 0 » A-λΙ is called the characteristic matrix Non-trivial solutions exist if and only if: » This is called the characteristic equation Characteristic polynomial » nth-order polynomial in λ » Roots are the eigenvalues {λ1, λ2, …, λn} 0)det( 21 22221 11211 = − − − =− λ λ λ λ nnnn n n aaa aaa aaa     IA
5. Eigenvalue Example Characteristic matrix Characteristic equation Eigenvalues: λ1 = -5, λ2 = 2       −− − =      −      − =− λ λ λλ 43 21 10 01 43 21 IA 0103)3)(2()4)(1( 2 =−+=−−−−=− λλλλλIA
6. Eigenvalue Properties Eigenvalues of A and AT are equal Singular matrix has at least one zero eigenvalue Eigenvalues of A-1 : 1/λ1, 1/λ2, …, 1/λn Eigenvalues of diagonal and triangular matrices are equal to the diagonal elements Trace Determinant ∑= = n j jTr 1 )( λA ∏= = n j j 1 λA
7. Determining Eigenvectors First determine eigenvalues: {λ1, λ2, …, λn} Then determine eigenvector corresponding to each eigenvalue: Eigenvectors determined up to scalar multiple Distinct eigenvalues » Produce linearly independent eigenvectors Repeated eigenvalues » Produce linearly dependent eigenvectors » Procedure to determine eigenvectors more complex (see text) » Will demonstrate in Matlab 0)(0)( =−⇒=− kk xIAxIA λλ
8. Eigenvector Example Eigenvalues Determine eigenvectors: Ax = λx Eigenvector for λ1 = -5 Eigenvector for λ1 = 2       − =     − =⇒ =+ =+ 3 1 or 9487.0 3162.0 03 026 11 21 21 xx xx xx       =      =⇒ =− =+− 1 2 or 4472.0 8944.0 063 02 22 21 21 xx xx xx 2 5 43 21 2 1 = −=       − = λ λ A 0)4(3 02)1( 43 2 21 21 221 121 =+− =+− ⇒ =− =+ xx xx xxx xxx λ λ λ λ
9. Matlab Examples >> A=[ 1 2; 3 -4]; >> e=eig(A) e = 2 -5 >> [X,e] = eig(A) X = 0.8944 -0.3162 0.4472 0.9487 e = 2 0 0 -5 >> A=[2 5; 0 2]; >> e=eig(A) e = 2 2 >> [X,e]=eig(A) X = 1.0000 -1.0000 0 0.0000 e = 2 0 0 2
10. Vector Spaces Real vector space V » Set of all n-dimensional vectors with real elements » Often denoted Rn » Element of real vector space denoted Properties of a real vector space » Vector addition » Scalar multiplication V∈x 0aawvuwvu a0aabba =−+++=++ =++=+ )()()( aaaaa aababa =+=+ =+=+ 1)( )()()( kckc ckkcccc
11. Vector Spaces cont. Linearly independent vectors » Elements: » Linear combination: » Equation satisfied only for cj = 0 Basis » n-dimensional vector space V contains exactly n linearly independent vectors » Any n linearly independent vectors form a basis for V » Any element of V can be expressed as a linear combination of the basis vectors Example: unit basis vectors in R3 021 =+++ (m)(2)(1) aaa mccc  V∈(m)(2)(1) aaa ,,,            =           +           +           =++= 3 2 1 3213321 1 0 0 0 1 0 0 0 1 c c c cccccc )((2)(1) aaax
12. Inner Product Spaces Inner product Properties of an inner product space Two vectors with zero inner product are called orthogonal Relationship to vector norm » Euclidean norm » General norm » Unit vector: ||a|| = 1 ∑= +++==⋅== n k nnkk T babababa 1 2211),( bababa 0ifonlyandif0),(0),( ),(),( ),(),(),( 2121 ==≥ = +=+ aaaaa acca cbcacba qqqq 22 2 2 1),( n T aaa +++=== aaaaa babababa +≤+≤),(
13. Linear Transformation Properties of a linear operator F » Linear operator example: multiplication by a matrix » Nonlinear operator example: Euclidean norm Linear transformation Invertible transformation » Often called a coordinate transformation )()()()()( xxxvxv cFcFFFF =+=+ Axy A yx = ∈ ∈∈ tionTransforma Operator ,Elements xnm mn R RR yAx Axy Ayx 1 x tionTransformaInverse tionTransforma ,,Dimensions − = = ∈∈∈ nnnn RRR
14. Orthogonal Transformations Orthogonal matrix » A square matrix satisfying: AT = A-1 » Determinant has value +1 or -1 » Eigenvalues are real or complex conjugate pairs with absolute value of unity » A square matrix is orthonormal if: Orthogonal transformation » y = Ax where A is an orthogonal matrix » Preserves the inner product between any two vectors » The norm is also invariant to orthogonal transformation bavuAbvAau ⋅=⋅⇒== ,    = ≠ = kj kj k T j if1 if0 aa vbua ==
15. Similarity Transformations Eigenbasis » If a nxn matrix has n distinct eigenvalues, the eigenvectors form a basis for Rn » The eigenvectors of a symmetric matrix form an orthonormal basis for Rn » If a nxn matrix has repeated eigenvalues, the eigenvectors may not form a basis for Rn (see text) Similar matrices » Two nxn matrices are similar if there exists a nonsingular nxn matrix P such that: » Similar matrices have the same eigenvalues » If x is an eigenvector of A, then y = P-1 x is an eigenvector of the similar matrix APPA 1ˆ − =
16. Matrix Diagonalization Assume the nxn matrix A has an eigenbasis Form the nxn modal matrix X with the eigenvectors of A as column vectors: X = [x1, x2, …, xn] Then the similar matrix D = X-1 AX is diagonal with the eigenvalues of A as the diagonal elements Companion relation: XDX-1 = A             ==⇒             = − nnnnn n n aaa aaa aaa λ λ λ         00 00 00 2 1 1 21 22221 11211 AXXDA
17. Matrix Diagonalization Example [ ]      − =     −       − ==       −      −      − ==       − =      − ==       ==      − =−=      − = − − − 20 05 215 45 13 21 7 1 13 21 43 21 13 21 7 1 13 21 7 1 13 21 1 2 ,2 3 1 ,5 43 21 1 1 1 21 2211 AXXD AXXD XxxX xxA λλ
18. Matlab Example >> A=[-1 2 3; 4 -5 6; 7 8 -9]; >> [X,e]=eig(A) X = -0.5250 -0.6019 -0.1182 -0.5918 0.7045 -0.4929 -0.6116 0.3760 0.8620 e = 4.7494 0 0 0 -5.2152 0 0 0 -14.5343 >> D=inv(X)*A*X D = 4.7494 -0.0000 -0.0000 -0.0000 -5.2152 -0.0000 0.0000 -0.0000 -14.5343
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