Eigen values and eigen vectors engineering
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Eigen values and eigen vectors engineering
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mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors. Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function ...
Eigen values and eigen vectors engineering
- Linear algebra: matrix Eigen-value Problems Eng. Shubham Kumbhar Part 3
- Eigenvalue Problems 1. Eigenvalues and eigenvectors 2. Vector spaces 3. Linear transformations 4. Matrix diagonalization
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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 +≤+≤),(
- 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
- 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 ==
- 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 − =
- 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
- 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
- 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