Transpose and Adjoint

The transpose of a matrix turns the matrix sideways. Suppose A is an m * n matrix with real number entries. Then the transpose A is ann * m matrix, and the (i,j) element of A is the (j,i) element ofA. Very concrete.

The adjoint of a linear operator is a more abstract concept, though it's closely related. The matrix A is sometimes called the adjoint ofA. That may be fine, or it may cause confusion. This post will define the adjoint in a more general context, then come back to the context of matrices.

This post, and the next will be more abstract than usual. After indulging in a little pure math, I'll return soon to more tangible topics such as Morse code and barbershop quartets.

Dual spaces

Before we can define adjoints, we need to define dual spaces.

LetV be a vector space over a fieldF. You can think of F as or . ThenV* is the dual space ofV, the space of linear functionals on V, i.e. the vector space of functions fromV toF.

The distinction between a vector space and its dual seems artificial when the vector space is ⁿ. The dual space of ⁿ is isomorphic to ⁿ, and so the distinction between them can seem pedantic. It's easier to appreciate the distinction betweenV andV* when the two spaces are not isomorphic.

For example, letV be L³(), the space of functions f such that |f|³ has a finite Lebesgue integral. Then the dual space is L^3/2(). The difference between these spaces is not simply a matter of designation. There are functionsf such that the integral of |f|³ is finite but the integral of |f|^3/2 is not, and vice versa.

Adjoints

The adjoint of a linear operator T:V W is a linear operator T*: W* V* whereV* and W* are the dual spaces of V and W respectively. SoT* takesa linear function fromW to the field F, and returns a function fromV toF. How doesT* do this?

Given an elementw* ofW*, T*w*takes a vector v inV and maps it to F by

(T*w*)(v) =w*(Tv).

In other words, T* takes a functional w* onW and turns it into a function onV by mapping elements ofV over toW and then lettingw* act on them.

Note what this definition does not contain. There is no mention of inner products or bases or matrices.

The definition is valid over vector spaces that might not have an inner product or a basis. And this is not just a matter of perspective. It's not as if our space has a inner product but we choose to ignore it; we might be working over spaces where it is not possible to define an inner product or a basis, such as , the space of bounded sequences.

Since a matrix represents a linear operator with respect to some basis, you can't speak of a matrix representation of an operator on a space with no basis.

Bracket notation

For a vector spaceV over a fieldF, denote a function , that takes an element fromV and an element fromV* and returns an element ofF by applying the latter to the former. That is, v, v* is defined to be the action ofv* onv. This isnot an inner product, but the notation is meant to suggest a connection to inner products.

With this notation, we have

Tv, w* _W = v, T*w* _V

for allv inVand for allwinWby definition. This is the definition of T* in different notation. The subscripts on the brackets are meant to remind us that the left side of the equation is an element ofF obtained by applying an element ofW* to an element ofW, while the right side is an element ofF obtained by applying an element ofV* to an element of V.

Inner products

The development of adjoint above emphasized that there is not necessarily an inner product in sight. But if there are inner products onV andW, then we can define turn an element ofv into an element ofV* by associatingv with , v where now the bracketsdo denote an inner product.

Now we can write the definition of adjoint as

Tv, w_W = v, T*w _V

for allv inVand for allwinW. This definition is legitimate, but it's not natural in the technical sense that it depends on our choices of inner products and not just on the operator T and the spacesV andW. If we chose different inner products onV andW then the definition ofT* changes as well.

Back to matrices

We have defined the adjoint of a linear operator in a very general setting where there may be no matrices in sight. But now let's look at the case of T:V W whereV andW are finite dimensional vector spaces, either over or . (The difference between and will matter.) And lets definite inner products on V andW. This is always possible because they are finite dimensional.

How does a matrix representation ofT* correspond to a matrix representation ofT?

Real vector spaces

SupposeV andW are real vector spaces andA is a matrix representation of T:V W with respect to some choice of basis on each space. Suppose also that the bases forV* andW* are given by the duals of the bases forV andW. Then the matrix representation ofT* is the transpose ofA. You can verify this by showing that

Av, w_W = v, Aw _V

for allv inVand for allwinW.

The adjoint of A is simply the transpose ofA, subject to our choice of bases and inner products.

Complex vector spaces

Now consider the case where V andW are vector spaces over the complex numbers. Everything works as above, with one wrinkle. If A is the representation of T:V W with respect to a given basis, and we choose bases forV* andW* as above, then theconjugate of A is the matrix representation ofT*. The adjoint ofA is A*, the conjugate transpose ofA. As before, you can verify this by showing that

Av, w_W = v, A^*w _V

We have to take the conjugate of A because the inner product in the complex case requires taking a conjugate of one side.

Related posts

• The Fredholm alternative
• Adjunctions in category theory
• Applied functional analysis

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