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1.a) Let A be any non-singular real square matrix. Show that A^T A (A transform * A) is symmetric and positive definite.
b) Let A= the matrix 14 7 7 . Find the eigenvalues of A. (*show working. *no use of computer packages)
7 6 1
7 1 6
c) Using the answer to part b), or otherwise, show that the matrix B=A + pI where p is a positive constant and I the usual identity matrix, is both symmetric and positive definite.
Any help would be greatly appreciated.
caddy
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also....
Consider the transformation R^5 --> R^5 defined by
8x1 + 6x2 + 4x3 + 6x4 + 2x5
8x1 + 11x2 +5x3 + 7x4 + 2x5
T(x)= -16x1 - 17x2 - 9x3 - 13x4 - 4x5
-18x1 - 19x2 - 11x3 - 17x4 - 6x5
34x1 + 36x2 + 20x3 + 30x4 +10x5
a) Find a matrix A such T(x) = Ax for all x in R^5.
b) Use an appropriate computer package to find all the eigenvalues and eigenvectors of A.
(Mathcad???)
c) Using the answer to part b), find the range, kernel, rank and nullity of T.
Thanks,
cads
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