24. Consider the quadratic form Q(x) = -x² - 2y² - z² + 2xy-2yza) Find a symmetric matrix A such that Q(x) = XT AX for all in R³b) Determine the eigenvalues of matrix A in (a) abovec) Determine the nature of definiteness matrix Ad) Find the nature rank of Q(x)e) Find signature of Q(x)

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Answered: 24. Consider the quadratic form Q(x) =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1 [1 2 Q-2: Let A = 1 1 2 and B 3 1 13 5 a) Find A-'B", if exists. b) Find the matrix C such that (AC + B)" = A" + B.
Is the quadratic form A = | positive definite, negative definite, positive semi-definite, negative semi-definite, or indefinite? (Hint: the matrix is not symmetric, so the theorems by using either the leading principal minors or the eigenvalues do not apply. Write down the explicit form of the quadratic form, then plug in some special values for .)
7.3
d) e) and f) only
Q2) If the complex nx n matrix A(t) is continuous on an interval I, then the solution of the system' = A(t)y on the interval I form a vector space V of dimension n over the complex numbers.
Q3) Prove that if 1, 12 and 13 are distinct eigenvalues of a matrix A with correspond ing eigenvectors v1 ,v2 and v3 , respectiv e ly , then vị ,v2 and V3 are linearly independen t .
Prove that if the determinant of a square matrix "A" does not equal to zero, then zero can not be an eigen value of "A".
3. If A = 0 -1 0then the bases for the eigenspaces of A2 + 21 are: 2. a) ({, = 7, v, = (1,0,0), {A2 = 4, v (-1,0,1)}, and (A = 1, v, = (-1,6,0}) b) {{ =-3, v, = (-1,6,0}), (A = 3, v (1,0,0}}, and (A = 0, v, = (1,0,0}] c) [{ = 27, v, = (1,0,0},{22 = 6, v = (-1,0,1}}, and (As = 3, s =(-1,6,0}} d) {{2 = 2, v, ={-1,0,1}}. [A2 = 23, , = [1,0,0), and (As =-1,, {-1,6,0}}
5. Prove that the matrices A and A' have the same eigen values. [K Prove th prooteriotic root of a matrix iff the matrix
2. Prove Cayley-Hamilton Theorem for a diagonalizable matrix, A: Let A be diagonalizable and p(x) be its characteristic polynomial; i.e. value of A. p(A) : = 0 for A, eigen- Show that p(A) = 0.

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24. Consider the quadratic form Q(x) = -x² - 2y² - z² + 2xy - 2yz a) Find a symmetric matrix A such that Q(x) = XT AX for all in R³ b) Determine the eigenvalues of matrix A in (a) above c) Determine the nature of definiteness matrix A d) Find the nature rank of Q(x) e) Find signature of Q(x)