Question 11.1. The general quadratic form on R2 is defined as:а bQ(r1; #2) = | x1 x2]ax? + 2bx1.x2 + caс аx2Assume that a > 0 and complete the square to show that Q will always bepositive if both Leading Principal Minors (LPM) of the quadratic coefficientmatrix are larger than 0. (In other words: show that (x1; x2) = (0; 0) is aminimum of Q if the quadratic coefficient matrix is positive definite).1.2. For each of the following quadratic forms, determine formally whetherx = 0 is a minimum, a maximum or a saddlepoint.(a) Q = 10x + 2x% – x1x2.(b) Q = –xỉ – 2.x3 – 2x1x2 along the linear subspace r1 – x2 = 0.

We have to answer the following for the given information.

1.1. The general quadratic form on R² is defined as: a b X1 Q(r1; x2) = | x1 x2 ax² + 2bx1x2 + cx, с d X2 Assume that a > 0 and complete the square to show that Q will always be positive if both Leading Principal Minors (LPM) of the quadratic coefficient matrix are larger than 0. (In other words: show that (x1; x2) = (0; 0) is a %3D minimum of Q if the quadratic coefficient matrix is positive definite). 1.2. For each of the following quadratic forms, determine formally whether x = 0 is a minimum, a maximum or a saddlepoint. %3D (a) Q = 10x² + 2.x3 – x1x2. (b) Q = -xỉ – 2x% – 2x1x2 along the linear subspace x1 – x2 = 0.

1.1. The general quadratic form on R² is defined as: a b X1 Q(r1; x2) = | x1 x2 ax² + 2bx1x2 + cx, с d X2 Assume that a > 0 and complete the square to show that Q will always be positive if both Leading Principal Minors (LPM) of the quadratic coefficient matrix are larger than 0. (In other words: show that (x1; x2) = (0; 0) is a %3D minimum of Q if the quadratic coefficient matrix is positive definite). 1.2. For each of the following quadratic forms, determine formally whether x = 0 is a minimum, a maximum or a saddlepoint. %3D (a) Q = 10x² + 2.x3 – x1x2. (b) Q = -xỉ – 2x% – 2x1x2 along the linear subspace x1 – x2 = 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Needed to be solve in 10 mints
solve the question no d and e
Use Use the following set of equations to solve questions (7,8, :9, 10, 11, 12) based on Gauss Elimination direct method 2.1 + x2 – x3 = 2 (1) 3x1 – 2.x2 + x3 = 1 (2) a1- 2.x2 + 2x3 = -1 (3) %3D The value of row 2 (R2) in the augmented matrix (A| b) after • :zeroing elements (a21, a31) is [4/3- ,5/3,4/3-,0] [4/3,5/3-,7/3,0] None [4/3,5/3-,4/3-,0]
For each of the following quadratic forms, write down the symmetric coefficient matrix A such that Q = vAv¹ and V = [X₁ X₂ Xn]. (a) Q (x, y, z) = -x² + 5 xy + 2y² + 2xz+3yz-3 z², (b) Q (x, y) = 2 x² + 4 xy + 4y², (c) Q (x, y, z) = -5x² + 5xy-3y² + 5xz+yz +4z², (d) Q (w, x, y, z) = −w² + 2 x² + 2 wy+xy − 5 y² − 5 w z - 4 xz - 5 y z + 3 z², (e) Q (x, y) = -2x² + 5xy-4y², A = A || A = A = = || II
) Find the standard form f(x) = a(x – h)² + k а(х - h)² + k of the quadratic function that has a vertex at and an X 2 9. 3 -intercept of 2 f(x) =
For matrix A = 15 A. 12 c. (15 -203) 20 4). 3 4 if f(x) = x²-2X, what is the answer to f(A)? 15 B. (112 -230) -15 20 D. 12 23 12 23
While trying to invert A, [A I] is carried to [P Q] by row operations. Show that P=QA.
18.- Define the symmetric matrix corresponding to the following quadratic form: F(x, y, z)= x² + z²-2xy+xz - 2yz
Consider the following quadratic form on R²: 43 2 57 2 + -x7² 25 -X₂ 2 25 Q(x) = [₁] = 48 a) Determine whether Q is positive or negative (semi)definite. b) Find a change of variables x = Py that converts Q to a form with no cross-product terms. c) write Q in terms of the new variables y 1 and y2. In your formula, write 'y_1' and 'y_2' for y₁ and y2. Use the square root symbol 'V' where needed to give an exact value for your answer. [00₁] 0 0 1 00 42 +. a) The quadratic form is c) Q(x) = 0 25 -X1X2 Positive Definite Negative Definite Positive Semidefinite Negative Semidefinite Indefinite
The standard form of the quadratic function f(x) = ax2 + bx +c is غدر مجاب عليه بعد a) f(x) = a(x –k)² +h b) f(x) = a(x – h)² – k c) f(x) = a(x – h)² + k d) f(x) = k(x – h)² + a 1.00 j عيد هذا السؤال إختر أحد الخيارات (b) (d)