Kenneth Kuttler

430 CHAPTER 19. EIGENVALUES AND EIGENVECTORS

34. The set, V ≡ {(x,y,z) : 2x+3y− z = 0} is a subspace of R3. Find an orthonormalbasis for this subspace.

35. The two level surfaces, 2x+ 3y− z+w = 0 and 3x− y+ z+ 2w = 0 intersect in asubspace of R4, find a basis for this subspace. Next find an orthonormal basis forthis subspace.

36. Let A,B be a m×n matrices. Define an inner product on the set of real m×n matricesby

(A,B)F ≡ trace(ABT ) .

Show this is an inner product satisfying all the inner product axioms. Recall for M ann×n matrix, trace(M)≡ ∑

ni=1 Mii. The resulting norm, ||·||F is called the Frobenius

norm and it can be used to measure the distance between two matrices.

37. The trace of an n×n matrix M is defined as ∑i Mii. In other words it is the sum of theentries on the main diagonal. If A,B are n×n matrices, show trace(AB) = trace(BA).Now explain why if A = S−1BS it follows trace(A) = trace(B). Hint: For the firstpart, write these in terms of components of the matrices and it just falls out.

38. For U a matrix, a number will be called o(U) if it satisfies lim∥U∥→0o(U)∥U∥ = 0. Here

∥U∥ will be the Frobenius norm of U . Show that for U an n×n matrix, det(I +U) =1+ trace(U)+o(U). Explain why if a number is a product of more than one entryof U then it must be o(U) . For example, U12U23 would be o(U). Hint: This is trueobviously if n = 1. Suppose true for n− 1 and expand along last column and useinduction to get the result for n.

39. Next show that if F−1 exists, then

det(F +U)−det(F) = det(F) trace(F−1U

)+o(U)

Hint: Factor out F from F +U .

40. Let A(t) be an m× n matrix whose entries are differentiable functions of t. Thesymbol A′ (t) , means to replace each t dependent entry of A(t) with its derivative.Thus if

A(t) =(

sin t t2

t +1 ln(1+ t2

) ) , then A′ (t) =(

cos t 2t1 2 t

t2+1

)Let A(t) be an m×n matrix and let B(t) be an n× p matrix. Show the product rule.

(AB)′ (t) = A′ (t)B(t)+A(t)B′ (t)

Hint: Just use the entries of both sides and reduce to the usual product rule. That is,the i jth entry of (AB)′ (t) is ∑k

(AikBk j

)′(t) . Now use the product rule.

4303435.36.37.38.39.40.CHAPTER 19. EIGENVALUES AND EIGENVECTORSThe set, V = {(x,y,z) :2x+3y—z=0} is a subspace of IR. Find an orthonormalbasis for this subspace.The two level surfaces, 2x + 3y —z+w =0 and 3x—y+z+2w =0 intersect in asubspace of R*, find a basis for this subspace. Next find an orthonormal basis forthis subspace.Let A, B be am x n matrices. Define an inner product on the set of real m x n matricesby(A,B) - = trace (AB’) .Show this is an inner product satisfying all the inner product axioms. Recall for M annxn matrix, trace(M) = Y?_, Mj. The resulting norm, ||-||,- is called the Frobeniusnorm and it can be used to measure the distance between two matrices.The trace of an n x n matrix M is defined as )}; Mj;. In other words it is the sum of theentries on the main diagonal. If A, B are n x n matrices, show trace (AB) = trace (BA).Now explain why if A = S~'BS it follows trace (A) = trace(B). Hint: For the firstpart, write these in terms of components of the matrices and it just falls out.For U a matrix, a number will be called o (U) if it satisfies limyy-s0 wr = 0). Here||U || will be the Frobenius norm of U. Show that for U ann xn matrix, det (I+ U) =1+trace(U) + 0(U). Explain why if a number is a product of more than one entryof U then it must be o(U). For example, Uj2U23 would be o(U). Hint: This is trueobviously if n = 1. Suppose true for n — 1 and expand along last column and useinduction to get the result for n.Next show that if F~! exists, thendet (F +U) —det(F) = det (F) trace (F-'U) +0(U)Hint: Factor out F from F+U.Let A(t) be an m xn matrix whose entries are differentiable functions of t. Thesymbol A’ (t), means to replace each ¢ dependent entry of A(t) with its derivative.Thus if. 2_ f sint t 17, { cost — 2taw=( 2 inte) )> tena’ = ( 1 a )Let A(t) be an m x n matrix and let B(t) be an n x p matrix. Show the product rule.(AB)' (t) =A’ (t)B(t) +A) B'()Hint: Just use the entries of both sides and reduce to the usual product rule. That is,the ij” entry of (AB)’ (t) is Y; (AixBuj), (t) . Now use the product rule.