19 Trace, Symmetric and Exterior Algebras, Determinant

Recap: tensor products

Last time: the tensor product of two vector spaces. is a vector space together with a bilinear map The elements are the pure tensors; general elements of are linear combinations of these. The defining property is:

bilinear maps linear maps , with on pure tensors, extended to general elements by linearity.

We saw : on pure tensors, , a rank 1 linear map. Given a basis of , the dual basis of , and a basis of , which is represented by the matrix with in position and in all other entries.

The trace of a linear operator

We can now properly define the trace of a linear operator!

In “ordinary” linear algebra classes, one defines the trace of an matrix to be , the sum of the diagonal entries. Then, noting that we have , and so the trace of is defined to be the trace of in any basis.

We could also try to define the trace via eigenvalues and their multiplicities, over an algebraically closed field: in a basis where is triangular it is manifest that .

We can do better (conceptually) by using and the contraction linear map . Namely, there is a natural bilinear pairing and it determines : on pure tensors , extended by linearity: .

This is indeed equivalent to the usual definition: choosing a basis and the dual basis , which matches the trace of the matrix with a single entry in position . So by linearity, where corresponds to the matrix .

Tensor products of several spaces

The tensor product can be defined as above: either using bases of , or as a quotient of a universal vector space by relations, or via the universal property for multilinear maps:

There is a multilinear map such that for every vector space and every multilinear map , there exists a unique such that .

In fact nothing new is happening, because .

We are primarily interested in the special case (by convention , ). Then we have bilinear maps for all , which taken together define a multiplication on the tensor algebra making it a noncommutative ring.

Symmetric algebra

Remember: we’ve seen that the space of bilinear forms decomposes into (symmetric and skew-symmetric bilinear forms). Equivalently: there is an involution (an automorphism such that ) taking , or on : .

has eigenvalues and eigenspaces , .

We can also do the same on higher tensor powers of or (the latter are multilinear forms). There is an action of the symmetric group on , i.e. each permutation defines a linear map and this defines a group homomorphism .

For example, , i.e. those with .

If , the symmetric part of a tensor can be determined by averaging: there is a linear map , given on pure tensors by

Still assuming , we could instead define as the quotient of by the subspace spanned by elements of the form , , . Explicitly (since transpositions generate ), by and the same for swapping other factors. This is different from (but isomorphic to) the previous definition.

To settle the question of which definition (as a quotient vs. a subspace of ) is better: the best definition is again by a universal property.

Recall comes with a multilinear map and is characterized by Now comes with a symmetric multilinear map and is characterized by

The product operations induce a product (using followed by averaging ). These combine to a product operation on called the symmetric algebra of .

is a commutative ring (and a vector space over : a -algebra). (Check: the product is still associative despite symmetrization by averaging: .)

(Simply: denote by and consider finite linear combinations of all these.)

More explicitly: if is a basis of , then any linear form on , , is of the form a degree 1 polynomial. Symmetric multilinear forms are, likewise, polynomials (with only degree terms): So: (where, by a sleight of hand, denotes the th coordinate of a vector in as a linear (degree 1 polynomial) function on , i.e. really this is another name for ).

Exterior algebra

Do the same thing for skew-symmetric, a.k.a. alternating, multilinear forms.

In characteristic zero, we can view as the image of the skew-symmetrization operator , (definition/notation). This is zero whenever for some , and so, by multilinearity, whenever are linearly dependent. Thus whenever !

Alternative definitions:

• the quotient of by the subspace spanned by and similarly for swapping any two factors.
• Or: is a vector space with an alternating multilinear map (, etc.), universal for alternating multilinear maps on .

If is a basis of , then , , form a basis of .

We have a product induced by the tensor algebra plus skew-symmetrization: This makes the exterior algebra into a (skew-commutative) ring, i.e. if , , then . (Check: .)

Notes not reached in this lecture, covered at the start of Lecture 20

Volume and determinant

If , then (if is a basis of , it is spanned by ).

A choice of isomorphism is determined by the data of a volume form , , i.e. a nondegenerate alternating multilinear map Think of this as: the signed volume of the parallelepiped with edge vectors is naturally , which becomes a scalar given .

Example 19.6. In a real inner product space with orthonormal basis , the natural volume form is , so (the volume of the unit cube is ). (Warning: reordering the basis gives ... orientation!)

Using the basis to identify , with for each : the determinant of an matrix!

Recall that the determinant of a matrix is is the only quantity which is

• multilinear in the columns of the matrix,
• alternating (swapping two columns gives ),
• .

Even though the notion of determinant/volume of vectors requires a choice of volume form (an isomorphism ), the notion of determinant of a linear operator requires no such choice!

• Usual definition: given , define , in any basis, using (the usual proof is painfully explicit), so under change of basis .
• Better definition: the exterior power is a functor, so induces a linear operator (explicitly, ). But , and any linear operator on a 1-dimensional vector space is a scalar multiple of . So define such that (This expresses the fact that scales the volume of parallelepipeds in by a factor of , without having to choose to measure those volumes.)

Using this definition of the determinant via , independence of the choice of basis is manifest, and so is the fact that !