Determinants

It is, of course, possible to generalize the considerations of the preceding section to multilinear forms and multiple tensor products. Instead of entering into that part of multilinear algebra, we proceed in a different direction; we go directly after determinants.

Suppose that is a linear transformation on an -dimensional vector space and let be an alternating -linear form on . If we write for the function defined by then is an alternating -linear form on , and, in fact, is a linear transformation on the space of such forms. Since (see Section: Alternating forms of maximal degree ) that space is one-dimensional, it follows that is equal to multiplication by an appropriate scalar. In other words, there exists a scalar such that for every alternating -linear form . By this somewhat roundabout procedure (from to to ) we have associated a uniquely determined scalar with every linear transformation on ; we call the determinant of , and we write . Observe that is neither a scalar nor a transformation, but a function that associates a scalar with each linear transformation.

Our immediate purpose is to study the function . We begin by finding the determinants of the simplest linear transformations, that is, the multiplications by scalars. If for every in , then for every alternating -linear form ; it follows that . We note, in particular, that and .

Next we ask about the multiplicative properties of . Suppose that and are linear transformations on , and write . If is an alternating -linear form, then so that . Since and it follows that (The values of are scalars, and therefore commute with each other.)

A linear transformation is called singular if and non-singular otherwise. Our next result is that is invertible if and only if it is non-singular. Indeed, if is invertible, then and therefore . Suppose, on the other hand, that . If is a basis in , and if is a non-zero alternating -linear form on , then by Section: Alternating forms , Theorem 3. This implies, by Section: Alternating forms , Theorem 2, that the set is linearly independent (and therefore a basis); from this, in turn, we infer that is invertible.

In the classical literature determinant is defined as a function of matrices (not linear transformations); we are now in a position to make contact with that approach. We shall derive an expression for in terms of the elements of the matrix corresponding to in some coordinate system . Let be a non-zero alternating -linear form; we know that If we replace each in the right side of (1) by and expand the result by multilinearity, we obtain a long linear combination of terms such as , where each is one of the ’s. (Compare this part of the argument with the proof of Section: Alternating forms , Theorem 3.) If, in such a term, two of the ’s coincide, then, since is alternating, that term must vanish. If, on the other hand, all the ’s are distinct, then for some permutation , and, moreover, every permutation can occur in this way. The coefficient of the term is the product . Since ( Section: Alternating forms , Theorem 1) is skew symmetric, it follows that where the summation is extended over all permutations in . (Recall that , by Section: Alternating forms , Theorem 3, so that division by is legitimate.)

From this classical equation (2) we could derive many special properties of determinants by straightforward computation. Here is one example. If and are permutations (in ), then (since is also a permutation), it follows that the products and differ in the order of their factors only. If, for each , we take , and then alter each summand in (2) accordingly, we obtain (Note that and that the sum over all is the same as the sum over all .) Since this last sum is just like the sum in (2), except that appears in place of , it follows from an application of (2) to in place of that

Here is another useful fact about determinants. If is a subspace invariant under , if is the transformation considered on only, and if is the quotient transformation , then . This multiplicative relation holds if, in particular, is the direct sum of two transformations and . The proof can be based directly on the definition of determinants, or, alternatively, on the expansion obtained in the preceding paragraph.

If, for a fixed linear transformation , we write , then is a function of the scalar ; we assert that it is, in fact, a polynomial of degree in , and that the coefficient of is . For the proof we may use the notation of (1). It is easy to see that is a sum of terms such as , where for exactly values of and for the remaining values of ( ). The polynomial is called the characteristic polynomial of ; the equation , that is, , is the characteristic equation of . The roots of the characteristic equation of (that is, the scalars such that ) are called the characteristic roots of .

EXERCISES