TRnxn Rnxn such that TAAT det T Rn Times n rightarrow Rn Tim

T:R^nxn--> R^nxn such that T(A)=A^T det=?

T: R^n Times n rightarrow R^n Timesn such that T (A) = A^T det T = ?

Solution

lets compute the eigen values of T .

T has an eigen value 1 with multiplicity (n+(n^2-n)/2)

consider vectors with 1 on the diagonal and all other elements zero . these are eigen vectors with eigen value 1 -total n

consider vectors with 1 on a fixed place other than diagonal and 1 at its transposed position  these are eigen vectors with eigen value 1- total (n^2-n)/2

T has eigen value -1 with multiplicity (n^2-n)/2 = n(n-1)/2

consider vectors with 1 on a fixed place other than diagonal and -1 at its transposed position  these are eigen vectors with eigen value -1- total (n^2-n)/2

determinant is the product of eigen values = -1^(n(n-1)/2)