Consider A 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 0

Consider A = [0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 -1 0 0 0 0 1 0 0 0 1 0 -1 0 1 0 -1 0 1 0 -1 0], let T: R^6 rightarrow R^6 be given by multiplying by this matrix. The characteristic polynomial of T is char_T(x) = x^5(x - 1)(you don\'t have to verify it here). What are the eigenvalue of T? Calculate the dimensions of ker T, ker T^2, ker T^3, ker(T - I).

Solution

1) Eigen values are the roots of the characteristic polynomial

Hence eigen values are = 0 and 1.

2) Since eigen value =0 with multiplicity of 5, T matrix is singular and rank = no of rows - multiplicity of root 0

= 6-5 =1

rank of matrix = dimension = 1

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T^2 will have eigen values as 1^2 and 0^2 and hence the same roots

Dim of T^2 = 1

Similarly dimension of powers of T will be 1.