Transpose & Adjoint: review & example

introduction

Let me pick up an old problem. The mathematics comes from the two posts “transpose matrix and adjoint operator” part 1 & part 2. The problem itself comes from the schur’s lemma post.

Upon further reflection, I am going to change the problem a little bit. Do not expect to see the same answers as before.

I am also going to work it twice, assuming that we are given different information as our starting point, but I’ll do it for the very same problem.

As I have said, the appropriate question for an introduction to ABO blood groups is: Can your mother donate blood to you? Until you can answer that question, you’re missing something about blood groups.

This example is not that good, but it does tie together the following concepts and computations:

transpose matrix & adjoint operator 2

(

begin digression

just in case you’ve seen this in another form, let me make some connections. if you don’t recognize any of this digression, that’s ok. you can move along, there’s nothing to see here.

i am taking a linear operator , from a vector space V to the same vector space V. in fact, i have more than a vector space: V is an inner product space; i have a “dot product”. 

what i call the reciprocal basis is often called the dual basis. in fact, halmos calls it the dual basis. but that terminolgy is also associated specifically with the so-called dual space V* of linear functionals on V; in that case, the dual basis is a basis in V*. there can be a great deal of confusion here. the dual space V* can be defined without an inner product on V; the inner product on V can be defined without ever mentioning the dual space V*. but if we introduce both inner product and V*, then there is a natural isomorphism between elements of V and of V*. i have seen people think of the one-to-one relationship between elements of V and V* as an identity, and to confuse the inner product of two vectors in V with the effect of a linear functional on a vector. (worse, i have seen people assert that an inner product involves one element of V and one element of V*.

there is a one-to-one correspondence between my right shoe and my left, but they are not identical. isomorphism is not always identity.

here, i have a finite-dimensional vector space V with an inner product on it. i have two bases (original and new) on V, and i want to construct a third basis for V. i call it the reciprocal basis to emphasize that it is not a basis on V*.

end digression

)

let’s see how this plays out.