Idempotent and nilpotent matrix proofs

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• To show that a given matrix is idempotent

  The solution is detailed and well presented.

• Nilpotent of a Matrix

  the matrix is nilpotent.

• Linear Algebra with Cubic Roots

  Proof: If is a nonzero complex number and is a nilpotent matrix, then is also a nilpotent matrix because implies . From induction of question C, we know that has a cubic root if is a nilpotent matrix.

• Show that a given matrix is symmetric and idempotent

  10052 Show that a given matrix is symmetric and idempotent Let X be a txk matrix whose columns are linearly independent. Let M=I-X(X'X)^(-1)X'. Show that M is symmetric and idempotent. M is symmetric if M'=M.

• Idempotent

  14719 Idempotent linear transformation A linear transformation L:V->V is said to be idempotent if L dot L = L. If L is idempotent, show that there exists a basis S={a1,a2,...,an} for V such that L(ai)=ai for i= 1,2,...,r and L(aj) = 0v for j= r+1,...

• Eigenvalues for Linear Algebra Class.

  An nxn matrix A is said to be nilpotent if A^k = O (the zero matrix) for some positive integer k. Show that all the eigen values of a nilpotent matrix are O. Proof: Suppose A is a nilpotent matrix.

• Nilpotent transformation

  ,N^k-1(g) are linearly independent, and then (assuming V has dimension n) If N is nilpotent of index n, show that the set S= {g, N(g), N^2(g),...,N^n-1(g)}is a basis for V. Describe the matrix which represents N with respect to the basis S.

• An eigenvalues problem

  478953 Eigenvalues ** Please see the attached file for the complete problem description ** An n * n matrix A is said to be nilpotent if A^k = O for some positive integer k. Show that all the eigenvalues of a nilpotent matrix are equal to zero.

• Nilpotent Groups

  86419 Nilpotent Groups Let G = UT (n,F) be the set of the upper triangular n x n matrices with entries in a field F with p elements and 1's on the diagonal. The operation in G is matrix multiplication.