How Do Matrix Transpose and Adjoint Relate to the Original Matrix?

[nolanp2]

i'm trying to understand the relationship between a matrix and it's adjoint and transpose. I'm trying to develope some sort of intuition but can't figure out what these matrices actually represent in relation to the original matrix. can anyone help me out?

 

[HallsofIvy]

It might be a good idea to move away from matrices and look at the general picture. If T is a linear tranformation from an "inner product space" V (a vector space with an inner product, <u, v>_V, defined) to an inner product space U (with inner product <u,v>_U) then the "adjoint" of T, T^t, is defined as the linear transformation from U back to V such that, for all u in U, v in V, <Tv,u>_U= <v,T^tu>_V. Note that both Tv and u are in u so that inner product must be the U product while v and T^tu are in V and so that inner product must be the V product.

In particular, U and V are finite dimesional vector spaces over the real numbers, so that T and T^t can be represented by matrices with real entries, T^t is just the "transpose"- swapping rows for columns. If U and V are finite dimensional vector spaces over the complex numbers, and T and T^t can be represented as matrices with comples entries, then, because of the requirement that <u,v>= complex conjugate of <v,u>, T^t must be also take the complex conjugate of entries in T.