Question

Let A be a symmetric matrix of size 3 × 3 and consider the following quadratic...

Let A be a symmetric matrix of size 3 × 3 and consider the following quadratic function f(x) = x T Ax. Show that the gradient of f is: ∇xf(x) = 2Ax. A matrix is symmetric if Aij = Aji for all i and j.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let Q(t) be a matrix, function of time, t. Assume that Q is orthogonal for all...
Let Q(t) be a matrix, function of time, t. Assume that Q is orthogonal for all t. Show that if Q(t_0) = I (identity matrix) for some t_0 then d/dt (Q(t_0)) is skew-symmetric.
Let A be a given (3 × 3) matrix, and consider the equation Ax = c,...
Let A be a given (3 × 3) matrix, and consider the equation Ax = c, with c = [1 0 − 1 ]T . Suppose that the two vectors x1 =[ 1 2 3]T and x2 =[ 3 2 1] T are solutions to the above equation. (a) Find a vector v in N (A). (b) Using the result in part (a), find another solution to the equation Ax = c. (c) With the given information, what are the...
(a) For the following quadratic forms gi(x) write down the associated symmetric matrix Ai such that...
(a) For the following quadratic forms gi(x) write down the associated symmetric matrix Ai such that gi(x) = xT Aix. g1(x1, x2) = x21 − 2x1x2 + 4x2 g2(x1, x2) = 4x21 − 6x1x2 + x2 g3(x1, x2, x3) = 3x21 + 3x2 + 5x23 + 2x1x2 − 2x1x3 − 2x2x3 g4(x1, x2, x3) = −3x21 − x2 + 8x2x3 − 16x23 (b) Determine the definiteness of g1(x) and g3(x) using the method of eigenvalues. (c) Determine the definiteness of...
Problem 3.2 Let H ∈ Rn×n be symmetric and idempotent, hence a projection matrix. Let x...
Problem 3.2 Let H ∈ Rn×n be symmetric and idempotent, hence a projection matrix. Let x ∼ N(0,In). (a) Let σ > 0 be a positive number. Find the distribution of σx. (b) Let u = Hx and v = (I −H)x and find the joint distribution of (u,v). 1 (c) Someone claims that u and v are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ. (e) Assume that 1 ∈ Im(H) and find...
Let B = [ aij ] 20×17 be a matrix with real entries. Let x be...
Let B = [ aij ] 20×17 be a matrix with real entries. Let x be in R 17 , c be in R 20, and 0 be the vector with all zero entries. Show that each of the following statements implies the other. (a) Bx = 0 has only the trivial solution x = 0 n R 17, then (b) If Bx = c has a solution for some vector c in R 20, then the solution is unique.
a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1...
a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1 ≤ i, j ≤ n) and having the following interesting property: ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n Based on this information, prove that rank(A) < n. b) Let A ∈ R m×n be a matrix of rank r. Suppose there are right hand sides b for which Ax = b has no solution,...
Consider W as the set of all skew-symmetric matrices of size 3×3. Is it a vector...
Consider W as the set of all skew-symmetric matrices of size 3×3. Is it a vector space? If yes, then find its dimension and a basis.
Consider the following. f(x, y) = x/y,    P(4, 1),    u = 3 5  i + 4 5  j...
Consider the following. f(x, y) = x/y,    P(4, 1),    u = 3 5  i + 4 5  j (a) Find the gradient of f. (b) Evaluate the gradient at the point P. (c) Find the rate of change of f at P in the direction of the vector u.
n x n matrix A, where n >= 3. Select 3 statements from the invertible matrix...
n x n matrix A, where n >= 3. Select 3 statements from the invertible matrix theorem below and show that all 3 statements are true or false. Make sure to clearly explain and justify your work. A= -1 , 7, 9 7 , 7, 10 -3, -6, -4 The equation A has only the trivial solution. 5. The columns of A form a linearly independent set. 6. The linear transformation x → Ax is one-to-one. 7. The equation Ax...
Let X be the set {1, 2, 3}. a)For each function f in the set of...
Let X be the set {1, 2, 3}. a)For each function f in the set of functions from X to X, consider the relation that is the symmetric closure of the function f'. Let us call the set of these symmetric closures Y. List at least two elements of Y. b) Suppose R is some partial order on X. What is the smallest possible cardinality R could have? What is the largest?