Properties of permutation matrices
Shortcut for permutations
A permutation matrix permutes (changes orders of) rows of a matrix. The precise meaning of this statement is given in equation (1) below.
Partitioning the matrix into rows we have
Stacking these equations we get
By analogy with we denote the last matrix
Thus, pre-multiplication by transforms to
If we had proven the multiplication rule for determinants, we could have concluded from (1) that
As we know, changing places of two rows changes the sign of by -1. (2) tells us that permutation by changes the sign of by In the rigorous algebra course (2) is proved using the theory of permutations, without employing the multiplication rule for determinants.
I am going to call (2) a shortcut for permutations and use it without a proof. In general, I prefer to use such shortcuts, to see what is going on and bypass tedious proofs.
Other properties of permutation matrices
Exercise 1. Prove that the Definition 1 is equivalent to the following: A permutation matrix is defined by two conditions: a) all its columns are unit column-vectors and b) no two columns are equal.
Proof. Take the th column. It contains one unity (the one that comes from the th unit row-vector). It cannot contain more than one unity because all rows are different. Hence, the th column is a unit column-vector. Different columns are different unit vectors because otherwise some row would contain at least two unities and would not be a unit vector.
Exercise 2. Prove that a permutation matrix is an orthogonal matrix.
Proof. By Exercise 1 we can write a permutation matrix as a matrix of unit column-vectors:
which proves orthogonality. It follows that (be careful with this equation, it follows from multiplicativity of determinants which we have not derived from our axioms).