### Matrix calculus: Determinants

### Permutations

In order to be able to deal with higher-dimensional determinants, we briefly discuss the essentials needed on permutations.

PermutationA **permutation** of #\{1,\ldots,n\}# is a list of the numbers #1,\ldots,n# in a specific order. The permutation #\sigma# is considered as a map from #\{1,\ldots,n\}# to itself determined by #\sigma(i) = \sigma[i]#.

This means that the image of #i# under #\sigma# is the #i#-th number of the list #\sigma#. Thus #\sigma = \rv{1,2}# is the identity on #\{1,2\}# and #\sigma = \rv{2,1}# interchanges #1# and #2#. We denote this permutation by #(1,2)#.

More generally, we denote by #(i,j)# the permutation of #\{1,\ldots,n\}# interchanging #i# and #j# and fixing all other numbers. (So the notation does not specify the number of elements #n#.) It is called a **transposition**. We have #(i,j)=(j,i)#.

If #\sigma(i) = i#, then we say that #\sigma# **fixes** the number #i#; if this is not the case, then we say that #\sigma# **moves** #i#.

The **permutation matrix** #P_\sigma# corresponding to #\sigma# is the #(n\times n)#-matrix of which the #(\sigma(i),i)#-entries for #i=1,\ldots,n# are equal to #1# and all other entries are #0#.

We will need to be able to distinguish between products of even and odd length.

The sign of a permutation

- A map #\{1,\ldots,n\}\to\{1,\ldots,n\}# is a permutation of #\{1,\ldots,n\}# if and only if it is a bijective map.
- The number of permutations of #\{1,\ldots,n\}# is equal to #1\cdot 2\cdots (n-1)\cdot n#. This number is often denoted by #n!# and called #n#
**factorial**. - Every permutation can be written as a product of transpositions. (The composition of #0# permutations is defined to be the identity map #\rv{1,2,\ldots,n}#.)
- If a permutation can be written in two ways as a product of transpositions, then both products have even length or both products have odd length.

The **sign** of a permutation #\sigma# is #1# if #\sigma# is the product of an even number of transpositions and #-1# otherwise. It is denoted #\text{sg}(\sigma)#.

Since #\sigma# maps the element #1# onto #2#, the permutation #\sigma# is the product of the transposition #(1,2)# and the permutation #\tau=[1,3,5,2,4]# that fixes #1#. Next we write #\tau# as a product of a transposition of the form #(2,a)# for suitable #a# and a permutation fixing #1# and #2#, and so on. This way we can write #\left[ 2 , 3 , 5 , 1 , 4 \right] # as the product of at most #4# transpositions.

We find

\[\begin{array}{rcl}

\left[ 2 , 3 , 5 , 1 , 4 \right] &=& (1,2)\,[1,3,5,2,4]\\ &=& (1,2)\,(2,3)\,[1,2,5,3,4]\\ &=& (1,2)\,(2,3)\,(3,5)\,[1,2,3,5,4]\\ &=& (1,2)\,(2,3)\,(3,5)\,(4,5)\,[1,2,3,4,5] \\

&=& (1,2)(2,3)(3,5)(4,5)

\end{array}

\]

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