Matrices
by definition, an nxn matrix T = (tᵢⱼ) is upper triangular if tᵢⱼ = 0 whenever i > j. Prove the following: If T is upper triangular and nilpotent, then tᵢᵢ = 0 for all i.
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Matrices
by definition, an nxn matrix A is nilpotent. if there is an integer k>0 such that A^k = 0, the zero matrix. Prove the following: If A is nilpotent, then I+A is nonsingular. (Here I is the indentity matrix of size nxn.)
Matrix Problem
by definition, an nxn matrix A is unipotent. If I-A is nilpotent. Prove the following: If A is unipotent, then A is nonsingular.
1. (All students!) For matrices with special properties, it is possible to create special version...
1. (All students!) For matrices with special properties, it is possible to create special versions of Gauss elimination. Suppose matrix A (nxn) is symmetric (which means that A-A). Suppose also that A is positive definite; this means that the scalar = xTAx is always 20 for every vector x , and J-0 only if x = 0 In this case it can be shown that the usual Gauss elimination process, which effectively creates the factorization A LU, can be simplified...
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for...
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
(a) When V P(R) a the usual differentiation operator is nilpotent. (b) Any upper triangular matrix...
(a) When V P(R) a the usual differentiation operator is nilpotent. (b) Any upper triangular matrix with all zeros on the diagonal is nilpotent.
5. Let AE Maxn(C). Recall that A is said to be nilpo tent if there exists a positive integer k such that A 0. Prove...
5. Let AE Maxn(C). Recall that A is said to be nilpo tent if there exists a positive integer k such that A 0. Prove the following statements (a) If A is nilpotent, then A 0. (Hint: First show that if A is nilpotent, then the Jordan form of A is also nilpotent.) (b) If A is nilpotent, then tr(A) 0 (e) A is nilpotent if and only if the characteristic polynomial of A is (-1)"" (d) If A is...
linear algebra Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and...
linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
7. Consider the Theorem: Suppose A and B are two lower triangular matrices (Defined in 8...
7. Consider the Theorem: Suppose A and B are two lower triangular matrices (Defined in 8 3.1), of order n. Then, the product AB is also a lower triangular matrix. Likewise for upper triangular matrices. (We say that the set of lower triangular matrices, of order n, is closed under multiplication.) Prove this theorem, for n = 3, by multiplying the following two matri- ces: a1 0 0 A bi b 0 1 0 0 and B 2 0 21...
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition...
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent matrix and use that along with the power series definition of the matrix exponential to find eAt 2!
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent...
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such...
Problem 4. Give an example of a linear operator T on a
finite-dimensional vector space such that T is not nilpotent, but
zero is the only eigenvalue of T. Characterize all such
operators.
Problem 5. Let A be an n × n matrix whose characteristic
polynomial splits, γ be a
cycle of generalized eigenvectors corresponding to an
eigenvalue λ, and W be the subspace spanned
by γ. Define γ′ to be the ordered set obtained from γ by
reversing the...
1. (All students!) For matrices with special properties, it is possible to create special versions of Gauss elimination. Suppose matrix A (nxn) is symmetric (which means that A-A). Suppose also that A is positive definite; this means that the scalar = xTAx is always 20 for every vector x , and J-0 only if x = 0 In this case it can be shown that the usual Gauss elimination process, which effectively creates the factorization A LU, can be simplified...
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
(a) When V P(R) a the usual differentiation operator is nilpotent. (b) Any upper triangular matrix with all zeros on the diagonal is nilpotent.
5. Let AE Maxn(C). Recall that A is said to be nilpo tent if there exists a positive integer k such that A 0. Prove the following statements (a) If A is nilpotent, then A 0. (Hint: First show that if A is nilpotent, then the Jordan form of A is also nilpotent.) (b) If A is nilpotent, then tr(A) 0 (e) A is nilpotent if and only if the characteristic polynomial of A is (-1)"" (d) If A is...
linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
7. Consider the Theorem: Suppose A and B are two lower triangular matrices (Defined in 8 3.1), of order n. Then, the product AB is also a lower triangular matrix. Likewise for upper triangular matrices. (We say that the set of lower triangular matrices, of order n, is closed under multiplication.) Prove this theorem, for n = 3, by multiplying the following two matri- ces: a1 0 0 A bi b 0 1 0 0 and B 2 0 21...
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent matrix and use that along with the power series definition of the matrix exponential to find eAt 2!
5. Let A be the matrix, 0 1 2 3 0 0 1 2 A o 0 0 4 A is a nilpotent matrix. Look up the definition of a nilpotent...
Problem 4. Give an example of a linear operator T on a
finite-dimensional vector space such that T is not nilpotent, but
zero is the only eigenvalue of T. Characterize all such
operators.
Problem 5. Let A be an n × n matrix whose characteristic
polynomial splits, γ be a
cycle of generalized eigenvectors corresponding to an
eigenvalue λ, and W be the subspace spanned
by γ. Define γ′ to be the ordered set obtained from γ by
reversing the...
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