- #1

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,784

- 18

Recall that for an nxn matrix A, the (i,i)-minor of A is defined as [itex]M_{ij}(A)=detA(i|j)[/itex], where A(j|i) stands for the matrix (n-1)x(n-1) obtained from A by removing the ith line and jth column.

Also note that we can view det as a map from R^n x ... x R^n to R taking n vectors from R^n, staking them as the lines of a nxn matrix and taking the determinant of that. And it is a well known fact from linear algebra that this map is n-linear.

In the same way, we can view [itex]M_{ij}[/itex] as a function from R^n x ... x R^n to R, and it is in this context that I ask if the (i,j)-minor of a matrix is a multilinear map.

My book says that it is, but I find it odd that if one multiplies the ith component a_i of [itex]M_{ij}[/itex] by a constant c (for instance 0), one gets not [itex]cM_{ij}(a_1,...,a_i,..,a_n)[/itex] as one should, but rather [itex]M_{ij}(a_1,...,a_i,..,a_n)[/itex] because the ith line is not taken into acount altogether! In other words, [itex]M_{ij}(a_1,...,a_i,..,a_n)[/itex] is independant of a_i !

Am I right?

Also note that we can view det as a map from R^n x ... x R^n to R taking n vectors from R^n, staking them as the lines of a nxn matrix and taking the determinant of that. And it is a well known fact from linear algebra that this map is n-linear.

In the same way, we can view [itex]M_{ij}[/itex] as a function from R^n x ... x R^n to R, and it is in this context that I ask if the (i,j)-minor of a matrix is a multilinear map.

My book says that it is, but I find it odd that if one multiplies the ith component a_i of [itex]M_{ij}[/itex] by a constant c (for instance 0), one gets not [itex]cM_{ij}(a_1,...,a_i,..,a_n)[/itex] as one should, but rather [itex]M_{ij}(a_1,...,a_i,..,a_n)[/itex] because the ith line is not taken into acount altogether! In other words, [itex]M_{ij}(a_1,...,a_i,..,a_n)[/itex] is independant of a_i !

Am I right?

Last edited: