Feb 19

Multilinearity in columns

Multilinearity in columns

The axioms and properties of determinants established so far are asymmetric in that they are stated in terms of rows, while similar statements hold for columns. Among the most important properties is the fact that the determinant of A is a multilinear function of its rows.  Here we intend to establish multilinearity in columns.

Exercise 1. The determinant of A is a multilinear function of its columns.

Proof. To prove linearity in each column, it suffices to prove additivity and homogeneity. Let A,B,C be three matrices that have the same columns, except for the lth column. The relationship between the lth columns is specified by A^{(l)}=B^{(l)}+C^{(l)} or in terms of elements a_{il}=b_{il}+c_{il} for all i=1,...,n. Alternatively, in Leibniz' formula

(1) \det A=\sum_{j_{1},...,j_{n}:\det P_{j_{1},...,j_{n}}\neq  0}a_{1j_{1}}...a_{nj_{n}}\det P_{j_{1},...,j_{n}}

for all i such that j_i=l we have

(2) a_{ij_i}=b_{ij_i}+c_{ij_i}.

If \det P_{j_1,...,j_n}\neq 0, for a given set j_1,...,j_n there exists only one i such that j_i=l. Therefore by (2) we can continue (1) as

\det A=\sum_{j_{1},...,j_{n}:\det P_{j_{1},...,j_{n}}\neq  0}a_{1j_{1}}...(b_{ij_{i}}+c_{ij_{i}})a_{nj_{n}}\det  P_{j_{1},...,j_{n}}=\det B+\det C.

Homogeneity is proved similarly, using equations A^{(l)}=kB^{(l)} or in terms of elements a_{il}=kb_{il} for all i.

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