is indetinite. 4. Let A be an n × n symmetric matrix with eigenvalues 1 indefini
ID: 3195839 • Letter: I
Question
is indetinite. 4. Let A be an n × n symmetric matrix with eigenvalues 1 indefinite if and only if at least one eigenvalue is positive and at least one eigenvalue is negative. , ,An. Prove that A is ' 5. Let f : Rn R,·ER", UER". Defin f(x. + tv). Prove that. is a global minimizer of f if and only if 0 is a global minimizer of pu for every vE RT. (Here we are not assuming that any derivatives for f or py exist.) e a new function po : R R by you (t) 6. Suppose that A is an n × n matrix. (a) Prove that the matrix B = A + AT) is symmetric. (b) Prove that x-Ax > 0 for all x 0 if and only if B is positive definite. (c) For the matrix A-1 4 9 | , show that x-Ax > 0 for all x 0 in R2Explanation / Answer
6. (a)
* (AT)T=A
* (A+B)T = AT + BT
Use this on B=1/2 (A +AT) to find that BT = B, and thus B is symmetric.
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