UNIVERSIDAD NACIONAL DE COLOMBIA, SEDE MEDELLNALGEBRA LINEAL APLICADA

PROFESOR:CARLOS E. MEJA

Taller # 1

SECTIONS 1.2-1.3

1. Prove each of the following.

(a) A set of n linearly independent vectors in Rn is a basis for Rn .

(b) The set {e1,e2, ...,en} is a basis of Rn .

(c) A set of m vectors in Rn , where m > n is linearly dependent.(d) Any two bases in Rn have the same number of vectors.

(e) span{v1,v2, ...,vk } is a subspace or Rn , where span{v1,v2, ...,vk } is the set of lin-

ear combinations of the k vectors {v1,v2, ...,vk } from a vector space Rn .

(f) span{v1,v2, ...,vk } is the smallest subspace of Rn containing {v1,v2, ...,vk }.

2. Prove that if S = {s1, s2, ...sk } is an orthogonal set of nonzero vectors, then S is linearlyindependent.

3. Let S an m-dimensional subspace ofRn . Then prove that S has an orthonormal basis.(Hint: Let {v1,v2, ...,vk }) be a given basis of S. Define a set of vectors {uk } by:

u1 = v1v1uk+1 =

v k+1v k+1

1

where

v k+1 = vk+1(vTk+1u1

)u1(vTk+1u2

)u2

(vTk+1uk

)uk ,

k = 1,2, ...,m1Then show that {u1,u2, ...,um} is an orthonormal basis of S. This is the classicalGrahm-Schmidt process

9. Let A be mn. Then A has rank 1 iff A can be written as A = abT , where a and b arenonzero column vectors.

SECTIONS 1.4-1.6

23. (a) Show that the matrix

H = I 2uuT

uTuwhere u is a column vector, is otrhogonal. (The matrix H is called House holdermatrix.)

(b) Show that the matrix

J =(

c ss c

), where c2+ s2 = 1

is orthogonal. (The matrix J is called Givensmatrix )

(c) Prove that the product of fwo orthogonal matrices is an orthogonal matrix.

(d) Prove that a triangular matrix that is orthogonal is diagonal.

24. Let A and B be two symmetric matrices.

(a) Prove that (A+B) is symmetric.(b) Probe that AB is not necessarly symetric. Derive a condition under which AB is

symmetric.

(c) If A and B are symmetric positive definite, prove that (A+B) is positive defi-nite. Is also AB positive definite? Give reasons for your answer. When is (AB)symmetric positive definite?

26. Let A be a symmetric positive definite matrix and x be a nonzero n-vector. Prove thatA+xxT is positive definite.

27. Prove that a diagonally dominant matrix is nonsingular, and a diagonally dominantsymmetric matrix whit positive diagonal entries is positive definite.

29. Prove that a symmetric matrix A is positive definite iff A1 exists and is positive defi-nite.

30. Let A be an mn matrix (m n) having full rank. Then AT A is positive definite.

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31. Prove the following basic facts un the eigenvalues and eigenvectors.

(a) A matrix A is nonsingular iff A does not have a zero eigenvalue.(Hint: det(A)=12...n .)

(b) The eigenvalues of AT and A are the same.

(c) If two matrices have the same eigenvalues, they need not be similar (constructyour own example to show this).

(d) A symmetric matrix is positive definite iff all its eigenvalues are positive.

(e) The eigenvalues of a triangular matrix are its diagonal elements.

(f) The eigenvalues of an orthogonal matrix have moduli 1.

(g) Let A be a symmetric matrix an letQ be orthogonal such thatQT AQ is diagonal.Then show that the columns of Q are the eigenvectors of A

(h) The eigenvectors of a symmetric matrix can be chosen to de orthogonal.

32. Let A be a symmetric matrix whit eigenvalues 1,2, ...,n , andorthonormal eigen-vectors v1,v2, ...,vn . Then show that

A =1v1vT1 +2v2vT2 + +nvnvTn

34. What are the singular values of a symmetric matrix? What are the singular values of asymmetric positive definite matrix? Prove that a square matrix A is nonsingular iff ithas no zero singular value.

SECTIONS 1.7-1.8

37. Show that, if x and y aretwo vectors, then||x|| ||y || ||x y || ||x||+ ||y ||38. If x and y are two n-vectors, then prove that

(a)xT y ||x||2||y ||2 (Cauchy-Schwarz inequality)

(b) ||xyT ||2 = ||x||2||y ||239. Let x and y two orthogonal vectors; then prove that

||x+ y ||22 = ||x||22+||y ||22

43. (a) Prove that the vector lenght is preserved by orthogonal matrix multiplication.That is, if x Rn and Q Rnn be orthogonoal, then ||Qx||2 = ||x||2 (isometrylemma).

(b) Is the statement in part (a) true if 1 and infinity norms are used? Give reasonsfor your answer.

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46. Prove that the spectral norm of a symmetric matrix is the same as its spectral radius.

47. Let A Rnn and let x, y , and z, be n-vectors such that Ax = b and Ay = b+ z. Thenprove that

||z||2||A||2

||x y ||2 ||A1||2||z||2

(assuming that A1 exists).

50. Prove that ||AT A||2 = ||A||2251. Prove that

||AB ||F ||A||F ||B ||F||AB ||F ||A||2||B ||F

52. Let A = (a1, ...,an), where a j is the j th column of A. Then prove that

||A||2F =ni=1

||ai ||22

53. Prove that if A and A+E are both nonsingular, then(A+E)1 A1 EA1(A+E)1(Banach lemma). What is the implication of this result?

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