The two matrices are row equivalent if they have the same size, and there is a permutation matrix that can be used to turn one into the other.
A permutation matrix is an identity matrix of the same size as the original matrix. It has no effect on the value of the original matrix but it will change its order.
Here’s an example:
Let’s say we have two matrices A and B, where
A = [a 1 , a 2 ] and B = [b 1 , b 2 ].
We can compute the permutation matrix P(A):
P(A) = [1 0 0; 0 1 0]
Now, we can use P(B) to turn B into A:
P(B) * B = [1 0 0; 0 1 0] * [b 1 , b 2 ] = [a 1 , a 2 ]
What does it mean for two matrices to be row equivalent?
Two matrices are row equivalent if they have the same number of rows and columns, and the same number of nonzero entries in each row. That is, they are identical except that one matrix has its rows permuted. For example:
The first matrix above is not row equivalent to the second because of the different order of its rows. However, both have the same number of rows and columns (namely 4), and both have 4 nonzero entries (namely 1, 3, 2, 3) in each row. So they’re essentially identical except for their order. This means we can write them as follows:
We say that A is “row equivalent” to B if there exists an invertible matrix C such that A = C^(-1)B.
Last modified: October 11, 2022