In Lecture 13, we discussed matrix representations of linear transformations between finite-dimensional vector spaces. In this lecture, we consider linear transformations between finite-dimensional Euclidean spaces, and discuss the relationship between the scalar product and the matrix representation of linear transformations. Note that any vector space can be promoted to a Euclidean space by choosing a basis in and defining to be the unique scalar product on such that is orthonormal.
Let and be Euclidean spaces; by abuse of notation, we will denote the scalar product in each of these spaces by the same symbol . Let be an orthonormal basis in and let be an orthonormal basis in Let be a linear transformation.
Definition 1: The matrix elements of relative to the bases and are the scalar products
The reason the number is called a “matrix element” of is that this number is exactly the -element of the matrix of defined in Lecture 13. Indeed, if
where the last equality follows from the orthonormality of However, one can note that it is not actually necessary to assume that and are finite-dimensional in order for the matrix elements of to be well-defined. However, we will always make this assumption, and thus in more visual form, we have that
The connection between matrices and scalar products is often very useful for performing computations which would be much more annoying without the use of scalar products. A good example is change of basis for linear operators. The setup here is that so that and are two (possibly) different orthonormal bases of the same Euclidean space. Given an operator we would like to understand the relationship between the two matrices
which represent the operator relative to the bases and respectively. In order to do this, let us consider the linear operator uniquely defined by the equations
Why do these equations uniquely determine ? Because, for any we have
Let us observe that the operator we have defined is an automorphism of i.e. it has an inverse. Indeed, it is clear that the linear operator uniquely determined by the equations
is the inverse of Operators which transform orthonormal bases into orthonormal bases have a special name.
Definition 2: An operator is said to be an orthogonal operator if it preserves orthonormal bases: for any orthonormal basis in the set is again an orthonormal basis in .
Note that every orthogonal operator is invertible, since we can always define just as we did above. In particular, the operators we defined above by are orthogonal operators.
Proposition 1: An operator is orthogonal if and only if
Proof: Observe that, by linearity of and bilinearity of it is sufficient to prove the claim in the case that and for some where is an orthonormal basis of
Suppose that is an orthogonal operator. Let Then is an orthonormal basis of and consequently we have
Conversely, suppose that
We then have that so that is an orthonormal basis of and thus is an orthogonal operator.
Proposition 2: An operator is orthogonal if and only if it is invertible and
Proof: Suppose first that is orthogonal. Then, is invertible and is also orthonal, and hence for any we have
Conversely, suppose that is invertible and
Then, for any we have
whence is orthogonal by Proposition 1.
Now let us return to the problem that we were working on prior to our digression into the generalities of orthogonal operators, namely that of computing the relationship between the matrices . We have
where we used Proposition 2 to obtain the second equality. Thus, we have the matrix equation
where on the right hand side we are using the fact that
is an algebra isomorphism, as in Lecture 13, which means that the matrix representing a product of operators is the product of the matrices representing each operator individually. This relationship between is usually phrased as the statement that the matrix representing the operator in the “new” basis is obtained from the matrix representing in the “old” basis by “conjugating” it by the of the matrix by the matrix where is the orthogonal operator that transforms the old basis into the new basis.