Question 4 Let A-Σ, λίνα, be the spectral decomposition with positive eigenvalues λι, ··· > 0...
Question 3 Set Let A-Σ 1 λ¡ViuT be the spectral decomposition with positive eigenvalues λ1,···Ae > 0. Ak Prove the following properties: 1. AT İs symmetric and AT PAP is its spectral decomposition: 3, Denote A-2-(A*) . Then A 2 E 1 where Question 3 Set Let A-Σ 1 λ¡ViuT be the spectral decomposition with positive eigenvalues λ1,···Ae > 0. Ak Prove the following properties: 1. AT İs symmetric and AT PAP is its spectral decomposition: 3, Denote A-2-(A*) ....
Question 2 Suppose A e Rxk is symmetric matrix. Let be the spectral decomposition. If λι,.. .λk are al nonzero, show that where At Question 2 Suppose A e Rxk is symmetric matrix. Let be the spectral decomposition. If λι,.. .λk are al nonzero, show that where At
= Let cos(6) sin(0) B - sin() cos() and 0 << 27 (i) Calculate the eigenvalues of B. Hence prove that the modulus of the eigenvalues is equal to one. (ii) Calculate the eigenvectors of B.
IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .
Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det (A) > 0. (1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
-a21 a21 d2,2 Let A=| all with au, a22 > 0. Show that Ax〉 0 yx 0. Argue that the eigenvalues of A are positive However, since A is not symmetric, conclude that A is not positive definite.
Please prove this theorem.. Theorem 4. The PDF of the F-statistic defined in (14) is given by &f) -(m+n f>o, fso 0,
If X is a normal random variable with μ =-2 and σ = 3, and has probability density function and cumulative density function fx (z), FX (z), calculate . P(-3< X < 0) F(1/4
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.
4. Let {Sn, n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y max{Sk, 1 Sk n, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain.