Each of the following expressions defines a function \(D\) on the set of \(3 \times 3\) matrices over the field of real numbers. In which of these cases is \(D\) a 3-linear function?
\(D(A) = A_{11} + A_{22} + A_{33}\);
\(D(A) = (A_{11})^2 + 3A_{11}A_{22}\);
\(D(A) = A_{11}A_{12}A_{33}\);
\(D(A) = A_{13}A_{22}A_{32} + 5A_{12}A_{22}A_{32}\);
\(D(A) = 0\);
\(D(A) = 1\).
Solution
1. Definition of a 3-linear function:
1.1. A function \(D: M_3(\mathbb{R}) \to \mathbb{R}\) is 3-linear (with respect to rows) if it is linear in each row \(R_1, R_2, R_3\) separately: For each \(i \in \{1, 2, 3\}\), with the other two rows fixed, the map \(R_i \mapsto D(R_1, R_2, R_3)\) satisfies \(D(R_i + c R_i') = D(R_i) + c D(R_i')\).
definition of multilinear function on matrices.
1.2. In terms of matrix entries, a polynomial function in \(\{A_{ij}\}\) is 3-linear if and only if every monomial is of degree 1 in the entries of row 1, degree 1 in the entries of row 2, and degree 1 in the entries of row 3, or is the identically zero function.
multilinear polynomials are sums of multilinear monomials.
2. Evaluation of each case:
2.1. (a) \(D(A) = A_{11} + A_{22} + A_{33}\): Not 3-linear. Scaling row 1 by \(c \in \mathbb{R}\) gives \(D(c R_1, R_2, R_3) = c A_{11} + A_{22} + A_{33} \neq c D(A)\) in general.
failure of homogeneity in row 1.
2.2. (b) \(D(A) = (A_{11})^2 + 3A_{11}A_{22}\): Not 3-linear. Scaling row 1 by \(c\) gives \(D(c R_1, R_2, R_3) = c^2 (A_{11})^2 + 3c A_{11} A_{22} \neq c D(A)\) (non-linear in row 1, degree 0 in row 3).
quadratic in \(R_1\).
2.3. (c) \(D(A) = A_{11}A_{12}A_{33}\): Not 3-linear. This expression is quadratic in the entries of row 1 (\(A_{11}A_{12}\)) and degree 0 in row 2.
failure of linearity in \(R_1\) and \(R_2\).
2.4. (d) \(D(A) = A_{13}A_{22}A_{32} + 5A_{12}A_{22}A_{32} = (A_{13} + 5A_{12}) A_{22} A_{32}\): 3-linear.
In \(R_1\): \(R_1 \mapsto A_{13} + 5A_{12}\) is a linear functional on row 1.
In \(R_2\): \(R_2 \mapsto A_{22}\) is a linear functional on row 2.
In \(R_3\): \(R_3 \mapsto A_{32}\) is a linear functional on row 3. The product of three linear functionals on distinct rows is a 3-linear function.
product of linear forms on independent factor spaces.
2.5. (e) \(D(A) = 0\): 3-linear. The constant zero function satisfies \(0 + c \cdot 0 = 0\) in each row argument, so it is 3-linear.
zero map is multilinear.
2.6. (f) \(D(A) = 1\): Not 3-linear. Scaling any row by 0 gives \(D(0, R_2, R_3) = 1 \neq 0 = 0 \cdot D(R_1, R_2, R_3)\).
non-zero constant functions are not linear.
3. Conclusion: \(D\) is a 3-linear function precisely in cases (d) and (e). Q.E.D.
2.