Let P and Q be n × n permutation matrices and v ∈ R^n a vector. Under what
conditions does the equation P v = Qv imply that P = Q?
To determine the conditions under which the equation P*v = Q*v implies that P = Q, we need to understand the properties of permutation matrices and how they act on vectors.
A permutation matrix is a square matrix that only contains 0s and 1s, where each row and each column has exactly one 1, and the rest are 0s. It represents a permutation of the rows or columns of an identity matrix.
Let's assume that n is the size of the permutation matrices. The product P*v represents the result of applying the permutation represented by P to the vector v. Similarly, Q*v represents the result of applying the permutation represented by Q to the vector v.
To prove that P = Q, we need to show that they have the same permutation, meaning that they rearrange the elements of v in the same way.
Now, let's consider the conditions:
1. P*v = Q*v for all v:
If P*v = Q*v for any vector v, then it implies that the permutation represented by P is the same as the permutation represented by Q. In this case, P = Q.
2. P*v = Q*v for some v ≠ 0:
If P*v = Q*v for some non-zero vector v, it implies that the permutation matrices P and Q are similar or equivalent for at least one vector v. However, this does not necessarily mean that P = Q. There could be other vectors for which the equation does not hold.
Therefore, the condition under which the equation P*v = Q*v implies that P = Q is when P*v = Q*v for all vectors v.
To determine this, we can check if the permutation matrices P and Q have the same permutation. We can do this by comparing their row or column permutations. If the row (or column) permutations are the same, then P = Q. Otherwise, P ≠ Q.