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![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](https://image.slidesharecdn.com/20121202c-151203144225-lva1-app6892/75/Eigen-values-and-eigen-vectors-engineering-9-2048.jpg)
![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](https://image.slidesharecdn.com/20121202c-151203144225-lva1-app6892/75/Eigen-values-and-eigen-vectors-engineering-16-2048.jpg)
![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](https://image.slidesharecdn.com/20121202c-151203144225-lva1-app6892/75/Eigen-values-and-eigen-vectors-engineering-17-2048.jpg)
![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](https://image.slidesharecdn.com/20121202c-151203144225-lva1-app6892/75/Eigen-values-and-eigen-vectors-engineering-18-2048.jpg)
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Eigen values and eigen vectors engineering
- 1. Linear algebra: matrixEigen-value Problems Eng. Shubham Kumbhar Part 3
- 2. Eigenvalue Problems 1. Eigenvaluesand eigenvectors 2. Vector spaces 3. Linear transformations 4. Matrix diagonalization
- 3. The Eigenvalue Problem Considera 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 Characteristicequation 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 ofA 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 determineeigenvalues: {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 vectorspace 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. Linearlyindependent 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 Innerproduct 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 ofa 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 » Ifa 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 thenxn 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=[-12 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