Let Xi be the column vector in Rn with the ith coordinate equal to 1 and all oth

Question

Let Xi be the column vector in Rn with the ith coordinate equal to 1 and all other coordinates equal to zero. Thus for instance, in R3 we have x1 = (1 0 0), x2 = (0 1 0), x3 = (0 0 1). Show that if A is any n x n matrix, we have that Axi, where aii is the ith entry on the main diagonal of A. Use the result of problem 1 to show the following: A positive definite matrix must have only positive numbers on its main diagonal. A negative definite matrix must have only negative numbers on its main diagonal. A positive semi definite matrix must have only nonnegative numbers on its main diagonal. A negative semi definite matrix must have only nonpositive numbers on its main diagonal. If a symmetric matrix has both negative and positive numbers on its main diagonal, it must be indefinite. If a symmetric matrix has a nonzero determinant and at least one zero on its main diagonal, it must be indefinite. Show that parts (a)-(d) in problem 2 are "necessary but not sufficient" conditions, in other words, there are matrices with all positive numbers on their diagonal that are not positive definite, etc.

Explanation / Answer

Dropbox Link for the answer file:

https://dl.dropboxusercontent.com/u/41585969/chegg/matrix_chegg.docx

(Apart from 2(f), answered all the parts).

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