PROVIDE PROOF FOR ANSWERS - POINTS WILL NOT BE GIVEN OTHERWISE: PROVIDE PROOF FO
ID: 2962148 • Letter: P
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PROVIDE PROOF FOR ANSWERS - POINTS WILL NOT BE GIVEN OTHERWISE:
PROVIDE PROOF FOR ANSWERS - POINTS WILL NOT BE GIVEN OTHERWISE:
Denote the following statements as True or False For a square matrix A RnTimesn, if all eigenvalues have negative real parts, then the determinant of A is negative. For a square and symmetric matrix A RnTimesn, if all eigenvalues have negative real parts, then the determinant of A is negative. For a square matrix A RnTimesn, if the rows of A are linearly independent, then the homogenous problem (Ax = 0) has only the trivial solution. For a square matrix A RnTimesn, if one of the eigenvalues of A is equal to zero, then the homogenous problem (Ax = 0) has only the trivial solution. For a square matrix A RnTimesn, all solutions of the matrix equation xk+1 = Axk approach to zero if all eigenvalues of A have negative real parts.Explanation / Answer
1)False consider counter example -1 0
0 -2 have eigen values -1 and -2 with determinant 2.
2) False with same example
3) True. All rows are linealy independent. So matrix is full rank which implies all columns are linearly independent. Now let A=[a1 a2 a3 .. an] where ai's are column vectors of matrix A. Ax= sum xi * ai =0 . Now if we have non zero solution for xi then ai's become linearly dependent which contrdicts the given statement. Hence x can have only all zeros solution.
4) False. Every square matrix can be decomposed as A=TDT^-1 ( with jordan form , see http://en.wikipedia.org/wiki/Jordan_normal_form ). Now det(A)=det(T)*det(D)*det(T^-1). If one eigen value of A is zero then the almost-diagonal matrix D with eigenvalues of A has one all zero row ,hence det(D) =0. As det(A) =0 , A is not full rank which implies columns of A are linealy dependent. Hence by argument in previous question we can have non zero solution.
5) False. counter example -1 0
0 -1 now x(k+1)= -x(k) = .... = (-1)^k *x(1) which does not approach to zero although A has negetive eigen values.
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