spaces

We are now ready to proceed with multilinear algebra. The basic concept is that of multilinear form (or functional), an easy generalization of the concept of bilinear form. Suppose that \(\mathcal{V}_1, \ldots, \mathcal{V}_k\) are vector spaces (over the same field); a k-linear form ( \(k = 1, 2, 3, \ldots\) ) is a scalar-valued function on the direct sum \(\mathcal{V}_1 \oplus \cdots \oplus \mathcal{V}_k\) with the property that for each fixed value of any \(k-1\) arguments it depends linearly on the remaining argument. The \(1\) -linear forms are simply the linear functionals (on \(\mathcal{V}_1\) ), and the \(2\) -linear forms are the bilinear forms (on \(\mathcal{V}_1 \oplus \mathcal{V}_2\) ). The \(3\) -linear (or trilinear) forms are the scalar-valued functions \(w\) (on \(\mathcal{V}_1 \oplus \mathcal{V}_2 \oplus \mathcal{V}_3\) ) such that \[w(\alpha_1 x_1 + \alpha_2 x_2, y, z) = \alpha_1 w(x_1, y, z) + \alpha_2 w(x_2, y, z),\] and such that similar identities hold for \(w(x, \alpha_1 y_1 + \alpha_2 y_2, z)\) and \(w(x, y, \alpha_1 z_1 + \alpha_2 z_2)\) . A function that is \(k\) -linear for some \(k\) is called a multilinear form .

Much of the theory of bilinear forms extends easily to the multilinear case. Thus, for instance, if \(w_1\) and \(w_2\) are \(k\) -linear forms, if \(\alpha_1\) and \(\alpha_2\) are scalars, and if \(w\) is defined by \[w(x_1, \ldots, x_k) = \alpha_1 w_1(x_1, \ldots, x_k) + \alpha_2 w_2(x_1, \ldots, x_k)\] whenever \(x_i\) is in \(\mathcal{V}_i\) , \(i = 1, \ldots, k\) , then \(w\) is a \(k\) -linear form, denoted by \(\alpha_1 w_1 + \alpha_2 w_2\) . The set of all \(k\) -linear forms is a vector space with respect to this definition of the linear operations; the dimension of that vector space is the product \(n_1 \ldots n_k\) , where, of course, \(n_i\) is the dimension of \(\mathcal{V}_i\) . The proofs of all these statements are just like the proofs (in Section: Bilinear forms ) of the corresponding statements for the bilinear case. We could go on imitating the bilinear theory and, in particular, studying multiple tensor products. In order to hold our multilinear digression to a minimum, we shall proceed instead in a different, more special, and, for our purposes, more useful direction.

In what follows we shall restrict our attention to the case in which the \(k\) spaces \(\mathcal{V}_i\) are all equal to one and the same vector space, say, \(\mathcal{V}\) ; we shall assume that \(\mathcal{V}\) is finite-dimensional. In this case we shall call a “ \(k\) -linear form on \(\mathcal{V}_1 \oplus \cdots \oplus \mathcal{V}_k\) ” simply a “ \(k\) -linear form on \(\mathcal{V}\) ,” or, even more simply, a “ \(k\) -linear form”; the language is slightly inaccurate but, in context, completely unambiguous. If the dimension of \(\mathcal{V}\) is \(n\) , then the dimension of the vector space of all \(k\) -linear forms is \(n^k\) . The space \(\mathcal{V}\) and, of course, the dimension \(n\) will be held fixed throughout the following discussion.

The special character of the case we are studying enables us to apply a technique that is not universally available; the technique is to operate on \(k\) -linear forms by permutations in \(\mathcal{S}_k\) . If \(w\) is a \(k\) -linear form, and if \(\pi\) is in \(\mathcal{S}_k\) , we write \[\pi w(x_1, \ldots, x_k) = w(x_{\pi(1)}, \ldots, x_{\pi(k)})\] whenever \(x_1, \ldots, x_k\) are in \(\mathcal{V}\) . The function \(\pi w\) so defined is again a \(k\) -linear form. (The value of \(\pi w\) at \((x_1, \ldots, x_k)\) is more honestly denoted by \((\pi w)(x_1, \ldots, x_k)\) ; since, however, the simpler notation does not appear to lead to any confusion, we shall continue to use it.)

Using the way permutations act on \(k\) -linear forms, we can define some interesting sets of such forms. Thus, for instance, a \(k\) -linear form \(w\) is called symmetric if \(\pi w = w\) for every permutation \(\pi\) in \(\mathcal{S}_k\) . (Note that if \(k = 1\) , then this condition is trivially satisfied.) The set of all symmetric \(k\) -linear forms is a subspace of the space of all \(k\) -linear forms. Hence, in particular, the origin of that space, the \(k\) -linear form \(0\) , is symmetric. For a non-trivial example, suppose that \(k = 2\) , let \(y_1\) and \(y_2\) be linear functionals on \(\mathcal{V}\) , and write \[w(x_1, x_2) = y_1(x_1)y_2(x_2) + y_1(x_2)y_2(x_1).\] This procedure for constructing \(k\) -linear forms has useful generalizations. Thus, for instance, if \(1 \leq h < k \leq n\) , and if \(u\) is an \(h\) -linear form and \(v\) is a \((k - h)\) -linear form, then the equation \[w(x_1, \ldots, x_k) = u(x_1, \ldots, x_h) \cdot v(x_{h+1}, \ldots, x_k)\] defines a \(k\) -linear form \(w\) , which, in general, is not symmetric. A symmetric \(k\) -linear form can be obtained from \(w\) (or, for that matter, from any given \(k\) -linear form) by forming \(\sum \pi w\) , where the summation is extended over all permutations \(\pi\) in \(\mathcal{S}_k\) .

We shall not study symmetric \(k\) -linear forms any more. We introduced them here because they constitute a very natural class of functions definable in terms of permutations. We abandon them now in favor of another class of functions, which play a much greater role in the theory.