Question:
Matrix Representation of a Linear Operator on N-dimensional Vector Space
1) Let A be a linear operator on n-dimensional vector space V if An-1 ε ≠0 and An ε = 0. For some ε ∈ V,prove that matrix representation A of A with respect to a basis.
[ 0000..........0]
[ 1000..........0]
[ 0000..........0]
[ 0100..........0]
X=(x ,x ,................x ) is similar to the [ 0001..........0]
[. ]
[. ]
[. ]
[.............. 10]
A) We have to prove that , A , A ,................ A .Is linearly independent which is easy.
B)The book said that the eigenvalues of A are all zero. This I don't know why.
C) The book said that the matrix of A under the basis is the matrix with all (i, i+1)-entries 1 and 0 elsewhere. This I don't understand.
2)Prove that if and is continuous in and in this one we are studying inner product.
2) Prove that if A2 =A than A is similar to a diagonal matrix
There is the answer
1. the only possible eigenvalues are 1 and 0. this I get it
2. Theonly eigenvalues of A are 0 and 1 Hence A is similar to a diagonal matrix of the form
D = (1,1,1,1,.....,0,0,.......0) where the number of 1's is equal to the rank of A. This I don't know why.