Since the same transformation can often be represented by many different matrices, depending upon the basis chosen to describe them, the following questions can be raised:
What properties of a matrix are the same independent of the basis, being intrinsic properties of the transformation the matrices represent?
When do two matrices represent the same transformation with different bases?
There are actually several questions that can be raised for each of these, because we can be describing matrices whose elements are all real, or we can permit complex elements, and we can insist that we stick to bases that are orthonormal (the dot product of any basis vector with itself is 1 and its dot product with any other basis vector is 0) or allow more general bases including those with complex components.
The answers are a bit different depending on which context we consider, but they are fundamentally similar.
We will here consider real matrices and real orthonormal bases only.
A matrix which takes our original basis vectors into another orthonormal set of basis vectors is called an orthogonal matrix; its columns must be mutually orthogonal and have dot products 1 with themselves, since these columns must form an orthonormal basis.
These conditions mean that an orthogonal matrix has its transpose as its inverse! (The condition for two matrices to be inverses of one another is that the rows of one are orthogonal to the columns of the other, except that rows and columns with the same index have dot product one with one another.)
The next question we address is: what happens to a matrix M when an orthogonal transformation A is applied to the original basis vectors?
A transforms the initial basis to A's columns. We want to know what the matrix M does to these column vectors. That is exactly what the matrix MA does to the original column basis vectors. A takes them into the new basis vectors and M then transforms these into whatever it does to them.
However the product MA expresses what M does to the new basis vectors in terms of the old ones; its columns give the effect of M on the new basis vectors as linear combinations of the old basis vectors.
We want to re-express these columns as linear combinations of the new basis vectors.
How do we do this?
The easiest way to see is to look at what happens when M is the identity matrix, I. This is the matrix which takes any vector into itself. After the change of basis, it must still take any vector into itself, so it must still be the identity matrix.
But if M = I then MA is just IA or A itself and that is what I becomes in terms of the old basis That is columns of A tell what the new basis vectors look like in terms of the old ones.
To re-express I in terms of the new basis you must do something which takes AI back into I. The way to do this is to multiply on the left by A-1 which is AT.
We deduce that multiplying on the left by A-1 performs the desired re-expression for I and therefore for any matrix M. We conclude that in the new basis the matrix M becomes ATMA.
The transpose of a matrix is the matrix obtained by interchanging its rows and columns.
A matrix is symmetric if it the same after such a transformation.
We have just seen that an orthogonal transformation changes a matrix M to one of the form ATMA, where ATA = I, and the matrix A has columns given by the new orthonormal basis expressed in terms of the old basis.
A nice feature of such a transformation is that if M is symmetric, it stays symmetric after any such transformation.
Exercise 32.4 Prove this claim: that M is symmetric if and only if ATMA is symmetric.
This tells us that the only matrices that can possibly be made diagonal by such transformations are symmetric; since when they are diagonal they are trivially symmetric.
If a matrix is diagonal, its eigenvectors are the basis vectors. So we have shown that only symmetric matrices have real orthonormal bases.
On the other hand, any matrix that is symmetric can be made diagonal by an orthogonal transformation. Another way to say this is there is an orthonormal basis of real eigenvectors for every symmetric matrix. Proof of this claim
We have answered our first question: which matrices can be put in diagonal form by choosing a new orthonormal basis? The answer being any symmetric matrix. And the way to put a matrix in such form is to find its eigenvectors and choose an orthonormal set of these.
Our second question was: when will two such matrices be representations of the same transformation in a different basis. And the answer is, when their characteristic equations are the same, so that their eigenvalues are the same and have the same multiplicities.