By Exercise 1 we can write a permutation matrix as a matrix of unit column-vectors: Prove that a permutation matrix is an orthogonal matrix. 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.Įxercise 2. It cannot contain more than one unity because all rows are different. It contains one unity (the one that comes from the th unit row-vector). Prove that 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. Other properties of permutation matricesĮxercise 1. In general, I prefer to use such shortcuts, to see what is going on and bypass tedious proofs. I am going to call (2) a shortcut for permutations and use it without a proof. Now, let x be a permutation of the elements of x. If we had proven the multiplication rule for determinants, we could have concluded from (1) thatĪs 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. 1 Let M i, j f ( x i, x j) where x is an n-dimensional vector and f is some well-behaved function. Thus, pre-multiplication by transforms to numpy.shuffle and numpy.permutation seem to permute only the rows of the matrix (not the columns at the same time).

Partitioning the matrix into rows we haveīy analogy with we denote the last matrix But, I would like to know if there is something more efficient that does this. The precise meaning of this statement is given in equation (1) below. Properties of permutation matrices Shortcut for permutationsĪ permutation matrix permutes (changes orders of) rows of a matrix.

Permute a matrix