5. For a random vector X = (X1, ..., Xn), we define its covariance matrix to be the n × n matrix C with %3D Cij = E[(X; – X;)(X; – X;)), where X; = E[X;]. Show that a covariance matrix C is always positive semi-definite. Can we say that it is always positive definite? hint: An n x n symmetric matrix A is called positive semi-definite if xT Ax > 0 for all x E R". If the inequality holds as a strict inequality for all non-zero x, it is called positive definite.

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Q: Suppose C is the cofactor matrix of matrix A and A CT then O a JA| = 0.5 %3D Ob.A= 1 %3D Od. A=+1

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Q: 11.4.4. f(x1,x2, X3, X4, X5) = (T1eª2, x3e¬²4, x5e*1), a = (0, – In 6, 4, ln 2, 11). %3D

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Q: Please see image

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Q: For a random vector X = (X1,..., Xn), we define its covariance matrix to be the n x n matrix C with…

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Q: 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: Ra R¹ by T(x) = Ax?

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