4. (a) Find the symmetric matrix A associated with the quadratic form, q = 5x -...
2. The spectral decomposition theorem states that the eigenstates of any Hermitian matrix form an orthonormal basis for the linear space. Let us consider a real 3D space where a vector is denoted by a 3x1 column vector. Consider the symmetric matrix B-1 1 1 Show that the vectors 1,0, and1are eigenvectors of B, and find 0 their eigenvalues. Notice that these vectors are not orthogonal. (Of course they are not normalized but let's don't worry about it. You can...
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...
Find the symmetric matrix A associated with the given quadratic form 2x2-3y2+z2 - 4xz A=
Find the symmetric matrix A associated with the given quadratic form 2x2-3y2+z2 - 4xz A=
9. Find the symmetric matrix A associated with the given quadratic form. A. x +223 +63122 B. 2x2 – 3y2 + x2 - 4x2
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,
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
Theorem. Consider the quadratic form Q(x) = Ar where A is anxn symmetric matrix and A, and denote the largest and smallest eigenvalues of A, respectively. Then max Q(x) = 2 = max Q() = 1 and Q0.) = 1, where is any unit eige vector corre sponding to ii) in (r) and QU.) where is any unit eigen vector corresponding to do 1. - Find max Q(x) and min Q(x). 1) Q(1) = 3x + 43273 +673 ii) Q(z)...
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)...
Consider the linear system dc dt = 5x + 2.3333333333333y, x(0) = 4 dy dt = – 2y, y(0) = - 3 If the associated matrix has the form M= с Find the entries. a = Preview Preview b= C= Preview d= Preview Find the trace and determinant of M. Preview tr(M) = det(M) = Preview Find the eigenvalues 11, 12 of M, where li > 12. 21 = Preview 12 = Preview Let vi = [1, yı] be an...