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1. (20 points) Let X (Xi, X, Xs) be a real random vector, where X, are identically dis- tributed and independent (i...
1. (20 points) Let X (Xi, X, Xs) be a real random vector, where X, are identically dis- tributed and independent (ii.d.) zero-mean Gaussian real random variables. Consider the random vector Y given by where A is a 3 x 3 real matrix and b is a 3 x 1 real vector. Justify all your answers. (a) Find the covariance matrix Cx of x. (b) Find the mean vector EY] of Y (c) Express the covariance matrix Cy of Y...
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B=...
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B=...
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
Need help with c). Any help would be greatly appreciated Let A be a square matrix...
Need help with c). Any help would be greatly appreciated
Let A be a square matrix and b be a vector and consider the system Ax = b. Gaussian elimination changes Ax = b to Rx = C, where R is the reduced row-echelon form of A. The solutions to this system are of the form 21 ouw X=1 +11+z1 for any real numbers y and z. 1. Find Randc 2. The row operations taking A to R are the...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
0 1] 1 01. 0 1] [3 3 3] [1 2. Let uy = (0,0,0) and...
0 1] 1 01. 0 1] [3 3 3] [1 2. Let uy = (0,0,0) and Cy = 3 5 5. Let Y = XA and Z = XB. Finally, let A = 0 13 5 6 lo a) Find My. b) Find Cy. c) Find B so that Z1, Z2, and Z3 are uncorrelated. d) Find the resulting Cz.
3 -0.751 (X1,X2, X3) be jointly Gaussian with ux (1,-2,3) and Cx 1. Let X =...
3 -0.751 (X1,X2, X3) be jointly Gaussian with ux (1,-2,3) and Cx 1. Let X = 3 0.25 4 L-0.75 0.25 Hint: If a set of random variables (RVs) are jointly Gaussian, then any subset of those RVs are also jointly Gaussian. Similarly, adding constants to (or taking linear combinations of) jointly Gaussian RVs results in jointly Gaussian RVs. Using this property you can solve problem 1 without using integration. When appropriate, you may express your answer by saying that...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and ide...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
- Every 2 x 2 real matrix M = ( a b. cd determines a complex...
- Every 2 x 2 real matrix M = ( a b. cd determines a complex function fM3+ iy) = UM(x,y) + ium (+ iy), where real-valued functions um and vm are determined by the following equation. umu,y) UM(2,y) 3)= (*) (*) (a) Show that there are constants wį and W2 E C such that fm(z) = W12 + W22. What are these constants in terms of a, b, c, d? [8] (b) Determine an equivalent condition on M such...
(a) Let A be a real n x m matrix. (i) State what conditions on n...
(a) Let A be a real n x m matrix. (i) State what conditions on n and m, if any, are needed such that the matrix AAT exists. Justify your statement. (ii) Assuming that the matrix AA exists, find its size. (iii) Assuming that the matrix AAT exists, prove using index notation that all diagonal elements of AAT are positive or equal to zero. (iv) Let 12 5 -3 A= 3-4 2 Calculate (AAT) -- (show all your working). 2)
1. (20 points) Let X (Xi, X, Xs) be a real random vector, where X, are identically dis- tributed and independent (ii.d.) zero-mean Gaussian real random variables. Consider the random vector Y given by where A is a 3 x 3 real matrix and b is a 3 x 1 real vector. Justify all your answers. (a) Find the covariance matrix Cx of x. (b) Find the mean vector EY] of Y (c) Express the covariance matrix Cy of Y...
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
(b) Consider the matrix differential equation for the vector x(t) d dt - B2+ where B= (69) 4 10 5 -1 (i) Find a particular solution to the matrix differential equation. (ii) Evaluate exp(Bt). (iii) Find the general solution to the matrix differential equation. Express the general solution in terms of the components of the vector (0).
Need help with c). Any help would be greatly appreciated
Let A be a square matrix and b be a vector and consider the system Ax = b. Gaussian elimination changes Ax = b to Rx = C, where R is the reduced row-echelon form of A. The solutions to this system are of the form 21 ouw X=1 +11+z1 for any real numbers y and z. 1. Find Randc 2. The row operations taking A to R are the...
(a) Let A be a fixed mx n matrix. Let W := {x ER" : Ax = 0}. Prove that W is a subspace of R". (b) Consider the differential equation ty" – 3ty' + 4y = 0, t> 0. i. Let S represent the solution space of the differential equation. Is S a subspace of the vector space C?((0.00)), the set of all functions on the interval (0,0) having two continuous derivatives? Justify ii. Is the set {tº, Int}...
0 1] 1 01. 0 1] [3 3 3] [1 2. Let uy = (0,0,0) and Cy = 3 5 5. Let Y = XA and Z = XB. Finally, let A = 0 13 5 6 lo a) Find My. b) Find Cy. c) Find B so that Z1, Z2, and Z3 are uncorrelated. d) Find the resulting Cz.
3 -0.751 (X1,X2, X3) be jointly Gaussian with ux (1,-2,3) and Cx 1. Let X = 3 0.25 4 L-0.75 0.25 Hint: If a set of random variables (RVs) are jointly Gaussian, then any subset of those RVs are also jointly Gaussian. Similarly, adding constants to (or taking linear combinations of) jointly Gaussian RVs results in jointly Gaussian RVs. Using this property you can solve problem 1 without using integration. When appropriate, you may express your answer by saying that...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
- Every 2 x 2 real matrix M = ( a b. cd determines a complex function fM3+ iy) = UM(x,y) + ium (+ iy), where real-valued functions um and vm are determined by the following equation. umu,y) UM(2,y) 3)= (*) (*) (a) Show that there are constants wį and W2 E C such that fm(z) = W12 + W22. What are these constants in terms of a, b, c, d? [8] (b) Determine an equivalent condition on M such...
(a) Let A be a real n x m matrix. (i) State what conditions on n and m, if any, are needed such that the matrix AAT exists. Justify your statement. (ii) Assuming that the matrix AA exists, find its size. (iii) Assuming that the matrix AAT exists, prove using index notation that all diagonal elements of AAT are positive or equal to zero. (iv) Let 12 5 -3 A= 3-4 2 Calculate (AAT) -- (show all your working). 2)
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