Nilpotent matrix is a matrix whose diagonal are all 0. A m = 0(A is a nilpotent

Question

Nilpotent matrix is a matrix whose diagonal are all 0. Am = 0(A is a nilpotent matrix where m is apositive integer). Consider n x n A matrix, and choose the smallest munber m suchthat Am = 0. Pick a vector v in Rn such that Am - 1v=/= 0. Show that the vectors v, Av, A2v,......, Am -1v are linearly independent. Hint: Consider a relation c0v + c1Av +c2A2v +......+ cm - 1Am -1v = 0. Multiply both sides of the equation with Am -1 to show that c0= 0, c1= 0, and soon. Highest rating within 24hr. Thanks. Nilpotent matrix is a matrix whose diagonal are all 0. Am = 0(A is a nilpotent matrix where m is apositive integer). Consider n x n A matrix, and choose the smallest munber m suchthat Am = 0. Pick a vector v in Rn such that Am - 1v=/= 0. Show that the vectors v, Av, A2v,......, Am -1v are linearly independent. Hint: Consider a relation c0v + c1Av +c2A2v +......+ cm - 1Am -1v = 0. Multiply both sides of the equation with Am -1 to show that c0= 0, c1= 0, and soon. Highest rating within 24hr. Thanks.

Explanation / Answer

Nilpotent matrix is a matrix whose diagonal are all 0. Am = 0(A is a nilpotent matrix where m is a positive integer). Consider n x n A matrix, and choose the smallest munber m such that Am = 0. Pick a vector v in Rn such that Am - 1v =/= 0. Show that the vectors v, Av, A2v,......, Am - 1v are linearly independent.

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