PlanetMath: determinant as a multilinear mapping

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determinant as a multilinear mapping

(Theorem)

Let $\mathbf{M}= (M_{ij})$ be an $n\times n$ matrix with entries in a field . The matrix $\mathbf{M}$ is really the same thing as a list of column vectors of size . Consequently, the determinant operation may be regarded as a mapping

$\displaystyle \det:\overbrace{K^n\times\ldots\times K^n}^{n \mbox{ times}}\rightarrow K$

The determinant of a matrix $\mathbf{M}$ is then defined to be $\det(\mathbf{M}_1,\ldots,\mathbf{M}_n),$ where $\mathbf{M}_j\in K^n$ denotes the $j^{\text{th}}$ column of $\mathbf{M}$ .

Starting with the definition

$\displaystyle \det(\mathbf{M}_1,\ldots,\mathbf{M}_n) = \sum_{\pi\in S_n} \mathrm{sgn}(\pi) M_{1\pi_1} M_{2\pi_2}\cdots M_{n\pi_n}$

(1)

the following properties are easily established:

the determinant is multilinear;
the determinant is anti-symmetric;
the determinant of the identity matrix is .

These three properties uniquely characterize the determinant, and indeed can -- some would say should -- be used as the definition of the determinant operation.

Let us prove this. We proceed by representing elements of as linear combinations of

$\displaystyle \mathbf{e}_1=\begin{pmatrix}1\\ 0\\ 0\\ \vdots\\ 0\end{pmatrix},\... ...d \ldots\quad \mathbf{e}_n=\begin{pmatrix}0\\ 0\\ 0\\ \vdots\\ 1\end{pmatrix},$

the standard basis of

. Let $\mathbf{M}$ be an $n\times n$ matrix. The $j^{\text{th}}$ column is represented as $\sum_i M_{ij}\mathbf{e}_i$ ; whence using multilinearity

$\displaystyle \det(\mathbf{M})$	$\displaystyle = \det\left(\sum_i M_{i1}\mathbf{e}_i\,,\sum_i M_{i2}\mathbf{e}_i\,,\;\ldots\;,\sum_i M_{in} \mathbf{e}_i\right)$
	$\displaystyle =\sum_{i_1,\ldots,i_n=1}^n M_{i_11} M_{i_22} \cdots M_{i_n n} \det(\mathbf{e}_{i_1},\mathbf{e}_{i_2},\ldots,\mathbf{e}_{i_n})$

The anti-symmetry assumption implies that the expressions $\det(\mathbf{e}_{i_1},\mathbf{e}_{i_2},\ldots,\mathbf{e}_{i_n})$ vanish if any two of the indices $i_1,\ldots,i_n$ coincide. If all

indices are distinct,

$\displaystyle \det(\mathbf{e}_{i_1},\mathbf{e}_{i_2},\ldots,\mathbf{e}_{i_n}) = \pm \det(\mathbf{e}_1,\ldots,\mathbf{e}_n),$

the sign in the above expression being determined by the number of transpositions required to rearrange the list $(i_1,\ldots,i_n)$ into the list $(1,\ldots,n)$ . The sign is therefore the parity of the permutation $(i_1,\ldots,i_n)$ . Since we also assume that

$\displaystyle \det(\mathbf{e}_1,\ldots,\mathbf{e}_n)=1,$

we now recover the original definition (1).

"determinant as a multilinear mapping" is owned by rmilson.

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See Also: exterior algebra

This object's parent.

Attachments:: basis-free definition of determinant (Definition) by mps

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Cross-references: permutation, parity, transpositions, number, indices, vanish, expressions, implies, standard basis, linear combinations, identity matrix, anti-symmetric, multilinear, properties, column, mapping, operation, determinant, size, column vectors, field, matrix
There are 2 references to this entry.

This is version 2 of determinant as a multilinear mapping, born on 2002-11-14, modified 2006-01-11.
Object id is 3595, canonical name is DeterminantAsAMultilinearMapping.
Accessed 3298 times total.

Classification:

AMS MSC:

15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)

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