(b) Find all the values of te R for which T is positive definite. 2 -1...
(b) Find all the values of t E R for which T is positive definite. 2 -1 (i) T = -1 -1 t 2 ; 2 (ii) T = 3 + 4 0
EXAMPLE 1 (a) Find the derivative of r(t) = (3 + t4)1+ te-y + sin(40k. (b) Find the unit tangent vector at the point t0. SOLUTION (a) According to this theorem, we differentiate each component of r: t 45 cos (4t) r(t) + 3 (b) Since r(0)= and r(o) j+4k, the unit tangent vector at the point (3, 0, 0) is i+ 4k T(0) = L'(0)--
EXAMPLE 1 (a) Find the derivative of r(t) = (3 + t4)1+ te-y +...
QUESTION 1 Find r' (3) if r(t) = t3i+tj + tk A. 3i + 2+ 1k B. 18 i +6j+1 k C. 9 ii+ 3j+0k D. 27 i +6j+1 k QUESTION 2 Find r' (11/2) if r(t) = 2sin(t) i + 3cos(t)j OA. -2 i + Oj 01 – 3j c.2 i + 3j D. 01 + 1 j QUESTION 3 Evaluate S 3421- 4 të jdt A. 31 - 4j i cu B.31 +4j cli - 1j Da bit...
sint Find lit to Find lim r(t) where Ple)=(2 + 1)i + te tj+ sinkk where r
6.2.3 Let U be a complex vector space with a positive definite scalar product and S, T e L(U) self-adjoint and commutative, so T-T o S. (i) Prove the identity 11(S iT)(u)ll-llS(11 )11 2 + llT(11)112, 11 e U. (6.2.10) (ii) Show that S ± iT is invertible if either S or T is so. However, the converse is not true. (This is an extended version of Exercise 4.3.4.)
6.2.3 Let U be a complex vector space with a positive...
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Prove that this function defines a vector norm. Hint: Use the Cholesky decomposition. (b) Find an example of square matrices A an This shows that ρ(A) is not a norm. Note: there are very simple examples. d B such that ρ(A+B)>ρ(A) + ρ(8)
(a) Let S be a symmetric positive definite matrix and define a function | on R"...
Material:
8.3.2 Consider the matrix (1 2 3 A-2 3 1 (8.3.28) (i) Use (8.3.27) to find the dominant eigenvalue of A. (ii) Check to see that u-(1 , I , î ), is a positive eigenvector of A. Use 11 and Theorem 8.6 to find the dominant eigenvalue of A and confirm that this is exactly what was obtained in part 0) obtained in part (i) or(ii ii) Compute all the eigenvalues of A directly and confirm the result...
1. Let A(?) := 2 ? ? 2 , where ? is a parameter. Find the values
of ? for which the matrix A(?) is positive definite. Find the
values of ? for which the matrix A(?) is positive
semidefinite.
1. Let 2 where ? is a parameter. Find the values of ? for which the matrix A(3) is positive definite. Find the values of ? for which the matrix A(3) is positive semidefinite.
a and b
Using definitions, check whether the following matrices are positive definite or positive semidefinite: 1 . 1 (2) 4-61. 8-6]. c-[--B 11. --G 9. -- 1 2 0 0 (b) A= 0 1 0 0 0 1 2 37 2 4 6 3 6 0 B= -1 2 D= -1 2 -1 9 -4 2 1 -1
Hta11 2. Prove that for the (Hilbert) matrix is positive definite. i+j-1 i.j-1 Hnts: (Proceed from the definition to show that if a-(a a in n, then ar Ha>0 .a, is a nonzero vector a 1s a nonzero vector (ii)--= Í xi +j-2 ax (111) manipulate a' Ha into the integral of a positive function. i+ J
Hta11 2. Prove that for the (Hilbert) matrix is positive definite. i+j-1 i.j-1 Hnts: (Proceed from the definition to show that if a-(a...