References: edX online course MIT 8.05.1x Week 3.
Sheldon Axler (2015), Linear Algebra Done Right, 3rd edition, Springer. Chapter 3.
We’ve seen that the matrix representation of a linear operator depends on the basis we’ve chosen within a vector space . We now look at how the matrix representation changes if we change the basis. In what follows, we’ll consider two sets of basis vectors and and two operators and . Operator transforms the basis into the basis , while does the reverse. That is
for all . From this definition, we can see that and , since
Theorem 1 An operator (like or above) that transforms one set of basis vectors into another has the same matrix representation in both bases.
Proof: In matrix form, we have (remember we’re using the summation convention on repeated indices):
Note that the matrix elements depend on different bases in the two equations.
We can now operate with again, using 1, to get
Comparing the last line with 6, we see that
Since the matrix elements are just numbers, this means that the elements in the two matrices and are the same.
We could do the same analysis using the operator with the same result:
We can now turn to the matrix representations of a general operator in two different bases. In this case, can perform any linear transformation, so it doesn’t necessarily transform one set of basis vectors into another set of basis vectors. Consider first the case where operates on each set of basis vectors given above:
Unless is an operator like or above, in general . We can see how these two matrices are related by using operators and above to write
We don’t need to specify the basis for the or matrices since the matrices are the same in both bases as we just saw above. The last line is just the expansion of in terms of the basis. In the penultimate line, we see that the quantity in square brackets is the product of 3 matrices:
The required transformation is therefore
As a check, note that if or , we reclaim the result in the theorem above, namely that and .
Trace and determinant
The trace of a matrix is the sum of its diagonal elements, written as . A useful property of the trace is that
We can prove this by looking at the components. If then
The trace of is the sum of its diagonal elements, written as , so
From this we can generalize to the case of the trace of a product of any number of matrices and obtain the cyclic rule:
Going back to 23, we have
Thus the trace of any linear operator is invariant under a change of basis.
For the determinant, we have the results that the determinant of a product of matrices is equal to the product of the determinants, and the determinant of a matrix inverse is the reciprocal of the determinant of the original matrix. Therefore
Thus the determinant is also invariant under a change of basis.