Properties:
The result is always a column vector when post multiplying by a vector: A(mXn) * u(nX1) = v(mX1)
Taking weighted combinations of the rows of the matrix where the weights are determined by the elements of the vector.
result is always a row vector when pre multiplying by a vector: Transpose(u(mX1))* A(mXn) = v(1Xn)
Taking weighted combinations of the columns of the matrix where the weights are determined by the elements of the vector.
In Symmetic matrix multiplication the rows and columns are the same so from the weights perspective there is no difference as can see followed:
Symmetric matrix multiplication
A symmetric matrix is a matrix which all the of diagonol elements are equal to their corrosponding elements row/column.
Su = v –> Transpose(Su) = Transpose(v) –> Transpose(Su) = Transpose(v) –> Transpose(u)*S = Transpose(v)
Therefore the result is the same when pre multiplying or post multiplying except that it needs to be transposed.
But from the weights perspective there is no difference.
