4. (15') (a) Find the orthogonal change-of-variable y -5x +42122 – 2x2 has no cross-terms in...
4. (15') (a) Find the orthogonal change-of-variable y -501 + 42102-2-3 has no cross-terms in terms of y. Q7 so that the quadratic form Q(2) (b) Determine the geometric type of the curve c = can. Q() and sketch a graph as accurately as you (c) Determine if Q is positive or negative definite or indefinite or semi-positive/negative definite.
Find an orthogonal change of variables that eliminates the cross product terms in the quadratic form Q, and express Q in terms of the new variables. 7x{ + 6x2 + 5x3 - 4X1X2 + 4x2X3 A substitution x = Py that eliminates cross-product terms is X1 = o A substitution x = Py that eliminates cross-product terms is Xi = – -}y.+3y2– žys, x2 = - - Žy2+3y2 +3v3, x3 = - {yı+ {y2– žv3. The new quadratic form is...
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.
5. (a) State the Spectral Theorem for Symmetric matrices. (b) Write out the spectral factorization of =(22 (c) Using the change of variables x Qy, where Q is an appropriate orthogonal matric, express the quadratic form 5ri + 4r1t2 2 in terms of yi and y2. Describe and sketch the graph of the equation 5xi +4z1t2 +22 1 in the z1, z2 plane (d) Find the spectral decomposition of A.
5. (a) State the Spectral Theorem for Symmetric matrices. (b)...
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)...
# 1: Consider the following curves in R la) 1822-32 x y + 37 U2 100. l ) 2x2 + 6 x y + 2 y-100. 1c) x2 + 4 x y + 4 y2-10:0. Write them in normal form. Give the change of variables that does this. For example, in 1a) the orthonormal basis of eigenvectors are λί 5,V1 (2,1)'/V5 and λ2 = St ( 100. ) . That is, 45, ½ = (1,-2)t/V5.S ( 1/V 5-2/v/5 ) (V6,...
Find the directions in the xy-plane in which the function fixy) 15-2x2 -y2 has zero change at the point P(2,1,3) Express the directions in terms of unit vectors What is the unt vector in a direction of zero change with a positive x-component? Type exact answers, using radicals as needed.) What is the other unit vector in a direction of zero change? (Type exact anowers, using radicals as needed.) cess Ubrary Enter your answer in each of the answer boxes...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...