Determine the definiteness of the quadratic form q(11, 12) = 9z; - 42222 +633.
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....
4. (a) Find the symmetric matrix A associated with the quadratic form, q = 5x - 4.1112+5x3, and compute the eigenvalues X, and 12 and the associated normalized eigenvectors e, and e2 of A. (b) Use the result of Part (a) to determine the spectral decomposition for A PAP. 22), and y. . wal. Rewrite q = (c) Let x = Py, where P is in Part (b), x = ( 5x - 4x32 +503 in y-variables, yı and y2.
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.
Consider the quadratic form Q = -2u2 - 6v2 + 6uv. A. Find the eigenvalues of the discriminant of Q. B. Using the eigenvalues, check the “definitness” of the quadratic form. C. Using the eigenvalues, calculate the determinant of the discriminant.
10') Determine the range of a so that the quadratic form Q(x, y, z) = a(r+y+z)+2xy-2yz+2zx positive definite.
(6) Let Q(x) = 2x; -4.01.22-23. Make a change of variable that transforms the quadratic form Q(2) into a quadratic form with no cross-product term.
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)...
Determine whether or not the given equations are quadratic. If the resulting form is quadratic identify a, b, and c, with a > o, Otherwise, explain why the resulting from is not quadratic 2x(2x5)-72 7) (2T 3) 1 (3x-2) 2
= TQu (9) Determine the maximum and minimum values that the quadratic form f(v) 2 0] obtains on the unit circle x2 + y2 = 1, where Q -22 = [2
Reduce to canonical form the following quadratic forms on R 3 : a) Q(x) = x12 + x 22 + 3x32+ 4x1x2 + 2x1x3 + 2x2x3; b) Q(x) = 2x1x2 − 6x2x3 + 2x1x3. The form Q = a1y12+ a2y22 + · · · + anyn2 , where y1, y2, . . . , yn are new unknowns, are called canonical