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1 4 Linear transformations as a vector space 17 d) Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation below find it matrix a) T : R 2 R 3 defined by T (x, y) T =(x +2y, 2x 5y, 7y) T ; b) T : R 4 R 3 defined by T (,x 2,, ) T =( +x 2 + +,x 2, + 3x 2 +6 ) T ; c) T : P n P n, Tf(t) =f (t) (find the matrix with respect to the standard basis 1,t,t 2,,t n ); d) T : P n P n, Tf(t) =2f(t) +3f (t) 4f (t) (again with respect to the standard basis 1,t,t 2,,t n ) 34 Find 3 3 matrices representing the transformations of R 3 which: a) project every vector onto x-y plane; b) reflect every vector through x-y plane; c) rotate the x-y plane through 30,leavingz-axis alone 35 Let A be a linear transformation If z is the center of the straight interval [x, y], show that Az is the center of the interval [Ax, Ay] Hint: What does it mean that z is the center of the interval [x, y]? 4 Linear transformations as a vector space What operations can we perform with linear transformations? We can always multiply a linear transformation for a scalar, ie if we have a linear transformation T : V W and a scalar α we can define a new transformation αt by (αt )v = α(t v) v V It is easy to check that αt is also a linear transformation: (αt )(α 1 v 1 + α 2 v 2 )=α(t (α 1 v 1 + α 2 v 2 )) by the definition of αt = α(α 1 T v 1 + α 2 T v 2 ) by the linearity of T = α 1 αt v 1 + α 2 αt v 2 = α 1 (αt )v 1 + α 2 (αt )v 2 If T 1 and T 2 are linear transformations with the same domain and target space (T 1 : V W and T 2 : V W, or in short T 1,T 2 : V W ), then we can add these transformations, ie define a new transformation T =(T 1 + T 2 ):V W by (T 1 + T 2 )v = T 1 v + T 2 v v V

2 6 Invertible transformations and matrices Isomorphisms 29 Recalling the definition of a basis we get the following corollary of Theorem 67 Corollary 69 An m n matrix is invertible if and only if its columns form a basis in R m Exercises 61 Prove, that if A : V W is an isomorphism (ie an invertible linear transformation) and v 1, v 2,,v n is a basis in V,thenAv 1,Av 2,,Av n is a basis in W 62 Find all right inverses to the 1 2 matrix (row) A =(1, 1) Conclude from here that the row A is not left invertible 63 Find all left inverses of the column (1, 2, 3) T 64 Is the column (1, 2, 3) T right invertible? Justify 65 Find two matrices A and B that AB is invertible, but A and B are not Hint: square matrices A and B would not work Remark: It is easy to construct such A and B in the case when AB is a 1 1 matrix (a scalar) But can you get 2 2 matrix AB? 3 3? n n? 66 Suppose the product AB is invertible Show that A is right invertible and B is left invertible Hint: you can just write formulas for right and left inverses 67 Let A be n n matrix Prove that if A 2 = 0 then A is not invertible 68 Suppose AB = 0 for some non-zero matrix B Can A be invertible? Justify 69 Write matrices of the linear transformations T 1 and T 2 in R 5, defined as follows: T 1 interchanges the coordinates x 2 and of the vector x, and T 2 just adds to the coordinate x 2 a times the coordinate, and does not change other coordinates, ie T 1 x 2 = here a is some fixed number x 2, T 2 x 2 = x 2 + a Show that T 1 and T 2 are invertible transformations, and write the matrices of the inverses Hint: it may be simpler, if you first describe the inverse transformation, and then find its matrix, rather than trying to guess (or compute) the inverses of the matrices T 1, T Find the matrix of the rotation in R 3 through the angle α around the vector (1, 2, 3) T We assume that rotation is counterclockwise if we sit at the tip of the vector and looking at the origin You can present the answer as a product of several matrices: you don t have to perform the multiplication 611 Give examples of matrices (say 2 2) such that: ;

3 1 Main definitions Eigenvalues of a triangular matrix Computing eigenvalues is equivalent to finding roots of a characteristic polynomial of a matrix (or using some numerical method), which can be quite time consuming However, there is one particular case, when we can just read eigenvalues off the matrix Namely eigenvalues of a triangular matrix (counting multiplicities) are exactly the diagonal entries a 1,1,a 2,2,,a n,n By triangular here we mean either upper or lower triangular matrix Since a diagonal matrix is a particular case of a triangular matrix (it is both upper and lower triangular the eigenvalues of a diagonal matrix are its diagonal entries The proof of the statement about triangular matrices is trivial: we need to subtract λ from the diagonal entries of A, and use the fact that determinant of a triangular matrix is the product of its diagonal entries We get the characteristic polynomial det(a λi) =(a 1,1 λ)(a 2,2 λ) (a n,n λ) and its roots are exactly a 1,1,a 2,2,,a n,n Exercises 11 True or false: a) Every linear operator in an n-dimensional vector space has n distinct eigenvalues; b) If a matrix has one eigenvector, it has infinitely many eigenvectors; c) There exists a square real matrix with no real eigenvalues; d) There exists a square matrix with no (complex) eigenvectors; e) Similar matrices always have the same eigenvalues; f) Similar matrices always have the same eigenvectors; g) The sum of two eigenvectors of a matrix A is always an eigenvector; h) The sum of two eigenvectors of a matrix A corresponding to the same eigenvalue λ is always an eigenvector 12 Find characteristic polynomials, eigenvalues and eigenvectors of the following matrices: ,, Compute eigenvalues and eigenvectors of the rotation matrix cos α sin α sin α cos α Note, that the eigenvalues (and eigenvectors) do not need to be real

4 104 4 Introduction to spectral theory Assume the field is C (the complex numbers) False in general otherwise Same comment 14 Compute characteristic polynomials and eigenvalues of the following matrices: , 0 π , e 0, Do not expand the characteristic polynomials, leave them as products 15 Prove that eigenvalues (counting multiplicities) of a triangular matrix coincide with its diagonal entries 16 An operator A is called nilpotent if A k = 0 for some k Prove that if A is nilpotent, then σ(a) ={0} (ie that 0 is the only eigenvalue of A) 17 Show that characteristic polynomial of a block triangular matrix A, 0 B where A and B are square matrices, coincides with det(a λi)det(b λi) (Use Exercise 311 from Chapter 3) 18 Let v 1, v 2,,v n be a basis in a vector space V Assume also that the first k vectors v 1, v 2,,v k of the basis are eigenvectors of an operator A, corresponding to an eigenvalue λ (ie that Av j = λv j, j =1, 2,,k) Show that in this basis the matrix of the operator A has block triangular form λik 0 B where I k is k k identity matrix and B is some (n k) (n k) matrix 19 Use the two previous exercises to prove that geometric multiplicity of an eigenvalue cannot exceed its algebraic multiplicity 110 Prove that determinant of a matrix A is the product of its eigenvalues (counting multiplicities) Hint: first show that det(a λi) = (λ 1 λ)(λ 2 λ) (λ n λ), where λ 1, λ 2,,λ n are eigenvalues (counting multiplicities) Then compare the free terms (terms without λ) or plug in λ = 0 to get the conclusion 111 Prove that the trace of a matrix equals the sum of eigenvalues in three steps First, compute the coefficient of λ n 1 in the right side of the equality, det(a λi) =(λ 1 λ)(λ 2 λ) (λ n λ) Then show that det(a λi) can be represented as det(a λi) =(a 1,1 λ)(a 2,2 λ) (a n,n λ)+q(λ) where q(λ) is polynomial of degree at most n 2 coefficients of λ n 1 get the conclusion And finally, comparing the

5 112 4 Introduction to spectral theory Exercises 21 Let A be n n matrix True or false: a) A T has the same eigenvalues as A b) A T has the same eigenvectors as A c) If A is is diagonalizable, then so is A T Justify your conclusions 22 Let A be a square matrix with real entries, and let λ be its complex eigenvalue Suppose v =(v 1,v 2,,v n ) T is a corresponding eigenvector, Av = λv Prove that the λ is an eigenvalue of A and Av = λv Here v is the complex conjugate of the vector v, v := (v 1, v 2,,v n ) T 23 Let Find A 2004 by diagonalizing A A = Construct a matrix A with eigenvalues 1 and 3 and corresponding eigenvectors (1, 2) T and (1, 1) T Is such a matrix unique? 25 Diagonalize the following matrices, if possible: 4 2 a) b) c) (λ = 2 is one of the eigenvalues) Consider the matrix A = a) Find its eigenvalues Is it possible to find the eigenvalues without computing? b) Is this matrix diagonalizable? Find out without computing anything c) If the matrix is diagonalizable, diagonalize it 27 Diagonalize the matrix Find all square roots of the matrix 5 2 A = 3 0 ie find all matrices B such that B 2 = A Hint: Finding a square root of a diagonal matrix is easy You can leave your answer as a product

6 2 Diagonalization Let us recall that the famous Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, is defined as follows: we put ϕ 0 = 0, ϕ 1 = 1 and define ϕ n+2 = ϕ n+1 + ϕ n We want to find a formula for ϕ n Todothis a) Find a 2 2 matrix A such that ϕn+2 = A ϕ n+1 ϕn+1 Hint: Add the trivial equation ϕ n+1 = ϕ n+1 to the Fibonacci relation ϕ n+2 = ϕ n+1 + ϕ n b) Diagonalize A and find a formula for A n c) Noticing that ϕn+1 ϕ n ϕ n = A n ϕ1 ϕ 0 = A n 1 0 find a formula for ϕ n (You will need to compute an inverse and perform multiplication here) d) Show that the vector (ϕ n+1 /ϕ n, 1) T converges to an eigenvector of A What do you think, is it a coincidence? 210 Let A be a 5 5 matrix with 3 eigenvalues (not counting multiplicities) Suppose we know that one eigenspace is three-dimensional Can you say if A is diagonalizable? 211 Give an example of a 3 3 matrix which cannot be diagonalized After you constructed the matrix, can you make it generic, so no special structure of the matrix could be seen? 212 Let a matrix A satisfies A 5 = 0 Prove that A cannot be diagonalized More generally, any nilpotent matrix, ie a matrix satisfying A N = 0 for some N cannot be diagonalized 213 Eigenvalues of a transposition: a) Consider the transformation T in the space M 2 2 of 2 2 matrices, T (A) = A T Find all its eigenvalues and eigenvectors Is it possible to diagonalize this transformation? Hint: While it is possible to write a matrix of this linear transformation in some basis, compute characteristic polynomial, and so on, it is easier to find eigenvalues and eigenvectors directly from the definition b) Can you do the same problem but in the space of n n matrices? 214 Prove that two subspaces V 1 and V 2 are linearly independent if and only if V 1 V 2 = {0}

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