4. (15') (a) Find the orthogonal change-of-variable y -501 + 42102-2-3 has no cross-terms in terms...
4. (15') (a) Find the orthogonal change-of-variable y -5x +42122 – 2x2 has no cross-terms in terms of y. Qī so that the quadratic form QT) = (b) Determine the geometric type of the curve c = Q(7) and sketch a graph as accurately as you can. (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.
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.
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(a) Find the square roots of the complex number z -3 + j4, expressing your answer in the form a + jb. Hence find the roots for the quadratic equation: x2-x(1- 0 giving your answer in the form p+ q where p is a real number and q is a complex number. I7 marks] (b) Express: 3 + in the form ω-reje (r> 0, 0 which o is real and positive. θ < 2π). Hence find the smallest value of...
# 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,...
2, It is known that the quantitative relationship between the dependent variable Y and the independent variables X and 2 is: A. Make a Table and draw the graph of the relationship between Y and X when the numerical value of Z is: B. Make a Table and draw the graph of the relationship between Y and Z when the numerical value of X is: C. Make a Table of the value of the rate of change of Y with...
2, It is known that the quantitative relationship between the dependent variable Y and the independent variables X and 2 is: A. Make a Table and draw the graph of the relationship between Y and X when the numerical value of Z is: B. Make a Table and draw the graph of the relationship between Y and Z when the numerical value of X is: C. Make a Table of the value of the rate of change of Y with...