Question

Consider the transposition linear map T : Mn×n (R) → Mn×n(R) given by

T([mij])=[mji].
1. Show that λ=±1 are the only eigenvalues of T.

Answer 1

The eigenvalues of the transposition linear map are λ=±1 because they correspond to symmetric and skew-symmetric matrices in Mn×n(R), and no other scalar will satisfy the condition for all matrices in Mn×n(R).

Step-by-step explanation:

The student is asking to show that λ=±1 are the only eigenvalues for the transposition linear map T : Mn×n (R) → Mn×n(R) given by T([mij])=[mji]. To find the eigenvalues of T, we must consider a matrix [mij] from Mn×n(R) and a scalar λ such that T([mij]) = λ[mij]. For λ=1, the matrix is unchanged, which corresponds to all symmetric matrices. For λ=-1, the matrix is negated, which corresponds to all skew-symmetric matrices where mij = -mji. No other scalar will satisfy the eigenvalue equation for all matrices in Mn×n(R), hence only λ=±1 are the eigenvalues of T.