representative matrix of a linear transformation given a linear transformation: T : M

Joshua Foley 2022-07-16 Answered
representative matrix of a linear transformation
given a linear transformation: T : M n ( C ) M n ( C ), T ( A ) = A 2 A T , what is the
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Answers (1)

Nicolas Calhoun
Answered 2022-07-17 Author has 15 answers
Let A denote the subspace of symmetric (antisymmetric matrices). Denote by E i j the matrix all of whose coefficients are zero except in the entry at the intersection of the i-th line and the j-th column, equal to 1. Thus ( ( E i j )) is the canonical basis of V.
For any i < j we have s p a n ( E i j , E j i ) = s p a n ( S i j , A i j ) where S i j = E i j + E j i 2 and A i j = E i j E j i 2
The transpose map τ : A A T is easily seen to be diagonal in that basis : we have
τ E i i = E i i , τ S i j = S i j , τ A i j = A i j
Note that S is the eigenspace of τ corresponding to the eigenvalue 1. So
{ E i i | 1 i n } { S i j | 1 i < j n } is a basis of S, and we deduce d i m ( S ) = n ( n + 1 ) 2 <brSimilarly { S i j | 1 i < j n } is a basis of A, and we deduce d i m ( A ) = n ( n 1 ) 2
For A S we have A T = A so σ ( A ) = A.
For A A we have A T = A so σ ( A ) = 5 A
So the characteristic polynomial χ A of A is
χ A = ( X + 1 ) d i m ( S ) ( X 5 ) d i m ( A ) = ( X + 1 ) n ( n + 1 ) 2 ( X 5 ) n ( n + 1 ) 2 χ A = ( X + 1 ) d i m ( S ) ( X 5 ) d i m ( A ) = ( X + 1 ) n ( n + 1 ) 2 ( X 5 ) n ( n + 1 ) 2
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