Consider the quadratic form Q - 2u24u,u2 5u22 + 2uzu32u,u 8. Write the quadratic form with the help of a matrix, in the...
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.
Consider the quadratic form Q(1, 2, r)2r2r34rs. Write Q(, 2, 3) in the fornm Q(1, 2, z3)xAx for some matrix A to be found, where x-2 T3 Classify Q(x1, r2, r3) as positive definite, negative definite, positive semidefinite, negative semidefinite, or indefinite
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 point) Write the matrix of the quadratic form Q(x, y, z) = 2.02 + 3y2 – 2z2 + 2xy + x2 + 8yz. A=
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.
1. [8 marks] Write equation 7x{ +6x1x2 + 7x3 = 1 as a matrix-vector quadratic form, convert it to a canonical form and determine the type of a curve to which it corresponds. 2. [16 marks] Find the spectral matrix and the corresponding modal matrix for -5 0 157 B = 3 4 -9. Write down the formula that needs to be used to diago- -5 0 15 nalise matrix B, but do not perform matrix multiplications.
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
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)...
X=0 x = 1/2 x= L u U2 Uz (a) Trial solution for a 1-D quadratic elastic bar element can be written as follows: ū(x) = [N]{u} where, [N] = [N1 N2 N3] and {u} u2 13 1 and Ni L2 L2 [N] and {u} are known as interpolation function matrix and nodal displacement, respectively. (272 – 3L + L´), N= = (22- La), Ns = 12 (2=– LE) Derive the expression for element stiffness matrix, (Kelem) and element force...