Math 19b: Linear Algebra with Probability Oliver Knill, Spring 2011Lecture 29: EigenvectorsEigenvectorsAssume we know an eigenvalue λ. How do we compute the corresponding eigenvector?The eigenspace of an eigenvalue λ is defined to be the linear space of all eigenvectorsof A to the eigenvalue λ.The eigenspace is the kernel of A − λIn.Since we have computed the kernel a lot a lready, we know how to do that.The dimension of the eigenspace of λ is called the geometric multiplicity of λ.Remember that the multiplicity with which an eigenvalue a ppears is called the algebraic multi-plicity of λ:The algebraic multiplicity is larger or equal than the geometric multiplicity.Pro of. Let λ be the eigenvalue. Assume it has geometric multiplicity m. If v1, . . . , vmis a basisof the eigenspace Eµform the matrix S which contains these vectors in the first m columns. Fillthe other columns arbitrarily. Now B = S−1AS has the property that the first m columns areµe1, .., µem, where eiare the standard vectors. Because A and B are similar, they have the sameeigenvalues. Since B has m eigenvalues λ also A has this property and the algebraic multiplicityis ≥ m.You can remember this with an analogy: the geometric mean√ab of two numbers is smaller orequal to the algebraic mean (a + b)/2.1 Find the eigenvalues and eigenvectors of the matrixA =1 2 3 4 51 2 3 4 51 2 3 4 51 2 3 4 51 2 3 4 5This matrix has a large kernel. Row reduction indeed shows that the kernel is 4 dimensional.Because the algebraic multiplicity is larger or equal than the geometric multiplicity thereare 4 eigenvalues 0. We can also immediately get the last eigenvalue from the trace 15. Theeigenvalues of A are 0, 0, 0, 0, 15.2 Find the eigenvalues of B.B =101 2 3 4 51 102 3 4 51 2 103 4 51 2 3 104 51 2 3 4 105This matrix is A + 100I5where A is the matrix from the previous example. Note that ifBv = λv then (A + 100I5)v = λ + 100)v so t hat A, B have the same eigenvectors and theeigenvalues of B a re 100, 100, 100, 100, 115.3 Find the determinant of the previous matrix B. Solution: Since the determinant is theproduct of the eigenvalues, the determinant is 1004· 115.4 The shear"1 10 1#has the eigenvalue 1 with algebraic multiplicity 2. The kernel of A−I2="0 10 0#is spanned by"10#and the geometric multiplicity is 1.5 The matrix1 1 10 0 10 0 1has eigenvalue 1 with algebraic multiplicity 2 a nd the eigenvalue 0with multiplicity 1. Eigenvectors to the eigenvalue λ = 1 are in the kernel of A −1 which isthe kernel of0 1 10 −1 10 0 0and spanned by100. The geometric multiplicity is 1.If all eigenvalues are different, then all eigenvectors are linearly independent andall geometric and algebraic multiplicities are 1. The eigenvectors form t hen aneigenbasis.Pro of. If all are different, there is one of them λiwhich is different from 0. We use inductionwith resp ect to n and assume the result is true for n−1. Assume that in contrary the eigen-vectors are linearly dependent. We have vi=Pj6=iajvjand λivi= Avi= A(Pj6=iajvj) =Pj6=iajλjvjso that vi=Pj6=ibjvjwith bj= ajλj/λi. If t he eigenvalues are different, thenaj6= bjand by subtracting vi=Pj6=iajvjfrom vi=Pj6=ibjvj, we get 0 =Pj6=i(bj−aj)vj= 0.Now (n−1) eigenvectors of the n eigenvectors are linearly dependent. Now use the inductionassumption.Here is an other example of an eigenvector computation:6 Find all the eigenvalues and eigenvectors of the matrixB =0 1 0 00 0 1 00 0 0 11 0 0 0.Solution. The characteristic polynomial is λ4− 1. It has the roots 1, −1, i, −i. Instead ofcomputing the eigenvectors f or each eigenvalue, writev =1v2v3v4,and look at Bv = λv.Where are eigenvectors used: in class we will look at some applications: H¨uckel theory, orbitalsof the Hydrogen atom and Page rank. In all these cases, the eigenvectors have immediateinterpretations. We will t alk about page rank more when we deal with Markov processes.The page rank vector is an eigenvector to the Google matrix.These matrices can be huge. The google matrix is a n×n matrix where n is larger than 10 billion!11The book of Lanville and Meyher of 2006 gives 8 billion. This was 5 years ago.Homework due April 13, 20111 Find the eigenvectors of the ma trixA =1 2 12 4 −23 6 −5.2 a) Find the eigenvectors of A10, where A is the previous matrix.b) Find the eigenvectors o f AT, where A is the previous matrix.3 This is homework problem 40 in section 7.3 of the Bretscher book.Photos of theSwiss lakesin the text.The pollutionstory is fictionfortunately.The vector An(x)b gives pollution levels in the Silvaplana, Sils and St Moritz lake n weeksafter an oil spill. The matrix is A =0.7 0 00.1 0.6 00 0.2 0.8and b =10000is the initial pollutionlevel. Find a closed form solution for the pollution after n