Consider the quadratic form Q(1, 2, r)2r2r34rs. Write Q(, 2, 3) in the fornm Q(1, 2,...
1. (10 points) Consider quadratic form q ) = ? Aš where: 1 0 C A= -2 3 -2 T=Y -3 -4 -5 ܠܛ 2 (a) Find a symmetric matrix Q such that q(7) = 2 Q7. (b) Determine whether the quadratic form q is positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite.
Classify the quadratic form below. Then make a change of variable, x= Py, that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. -5x2 - 12x4x2 What is the most precise classification for the quadratic form? O A. Positive definite O B. Indefinite O C. Negative semidefinite OD. Positive semidefinite O E. Negative definite The new quadratic form is y'Dy=N.
Consider the quadratic form Q - 2u24u,u2 5u22 + 2uzu32u,u 8. Write the quadratic form with the help of a matrix, in the form XAX 9. Examine whether the "definitness" of this quadratic form 10. BONUS (1 point). Calculate the eigenvalues of the discriminant and use them to check your answer to question 9 Consider the quadratic form Q - 2u24u,u2 5u22 + 2uzu32u,u 8. Write the quadratic form with the help of a matrix, in the form XAX 9....
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that Q(X) = XTAX. (b) Find an orthogonal matrix Q so that the change of variables x = Qy transforms the quadratic for Q(x) into one with no cross-product terms, that is, diagonalize the quadratic form (x). Give the transformed quadratic form. (c) Find a vector x of length 1 at which Q(x) is maximized. (d)...
Quadratic forms arise often in geometry, physics and engineering and it is desirable to reduce these to canonical (standard) forms. Reduce the quadratic form:Q = 3x12 + 5 x22 + 3x32 - 2x2 x3 + 2x1 x3 - 2 x1 x2 to a standard form by making an approximate change in variables X = MY , where M is an orthogonal matrix. i) Write Q as X T AX . ii) Find the Eigenvalues of A. Is the quadratic form...
5.3.15 Consider the quadratic form tx In (5.3.21) 1) Find a symmetric matrix A E R(n, n) such that q(x)-x' Ax for (ii) Compute the eigenvalues of A to determine whether q or A is pos- r E R" itive definite,
1) Compute the gradient of f and the Hessian of f. 2) Is the Hessian positive semidefinite, positive definite, negative definite, negative semidefinite, or indefinite at the following points: (1, 1, 5, 0) and (1, 1, 5, 2) and (1, 1, 1, 2)? Let f (x1 , X2, X3, X4) X1 . X2-X3 . (11mP + 100x1 ex2+ รื่ 1+2)2 4-
[1 2 2 3 2 3 (a) Factorize the matrix A =| 2 | into elementary matrices. (b) Write the condition for positive and negative definite quadratic forms. Reduce the quadratic form q=4x7+3x2-x?+2x2x3-4x3x1+4x1x2 to the canonical form. Hence find rank, index and signature of q. Write down the corresponding equations of transformation.
Exercise 2 Consider the symmetric matrix A a13 23 012 a13 023 , the quadratic form .q(z) = z'Az, associated T2 T3 1. Show that for x = with the symmetric matrix A is 2. Using the result from question (1), find the matrix associated with the quadratic forms below. Assumed that x is in IR3
10') Determine the range of a so that the quadratic form Q(x, y, z) = a(r+y+z)+2xy-2yz+2zx positive definite.