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Section 46.2 Concepts

In Part III, we went to great lengths to determine the “simplest form” a square matrix could take. It turns out that essentially we were studying linear operators on finite-dimensional spaces and the best way to choose a single basis for the space so that the matrix of the operator has a particular form.

Subsection 46.2.1 Matrices of an operator

Suppose \(\funcdef{T}{V}{V}\) is a linear operator on a finite-dimensional vector space \(V\text{.}\) For a choice of basis \(\basisfont{B}\text{,}\) we may form the square matrix \(\matrixOf{T}{B}\) that represents the matrix and allows us to perform matrix calculations that are equivalent to computing outputs for the operator.

Suppose \(\basisfont{B}'\) is another choice of basis for \(V\text{.}\) The matrices \(\matrixOf{T}{B}\) and \(\matrixOf{T}{B'}\) should be related somehow, since they each correspond to the same linear operator on \(V\text{,}\) just relative to different bases.

Diagram of transformations \(\R^n \to V \to V \to \R^n\)
Figure 46.2.1. Illustration of a linear operator versus the matrix operator created by coordinate maps relative to two different choices of bases for the domain space.

To relate the two matrices, it would seem to make sense to work in a change of basis. According to our analysis in Subsection 45.3.3, we can change basis using the identity operator, but choosing different bases for domain and codomain versions of the vector space.

Diagram of a change of basis for the matrix of an operator
Figure 46.2.2. Illustration of a linear operator versus the matrix operator created by coordinate maps, incorporating a change of basis.

In the chain of transformations along the top leg of Figure 46.2.2, we can use Statement 1 of Proposition 44.5.3 to simplify

\begin{equation*} I_V T I_V = T \text{.} \end{equation*}

With this in mind, the composition along the bottom leg must be represented by \(\matrixOf{T}{B'}\text{,}\) since the two long arrows at top and bottom create a square equivalent to the \(T\) versus \(\matrixOf{T}{B'}\) square on the right in Figure 46.2.1. Yet we know how compositions work: the matrix of a composition is the product of the matrices (Statement 1 of Proposition 45.5.6). So we must have

\begin{equation*} \matrixOf{T}{B'} = \matrixOf{I_V T I_V}{B'} = \matrixOf{I_V}{B'B} \matrixOf{T}{B} \matrixOf{I_V}{BB'} \text{.} \end{equation*}

Applying the facts that the matrix of an identity operator is a transition matrix (Proposition 45.5.3) and that the inverse of a transition matrix represents the reverse change of basis (Statement 3 of Proposition 22.5.4), we obtain

\begin{gather} \matrixOf{T}{B'} = \matrixOf{I_V}{B'B} \matrixOf{T}{B} \matrixOf{I_V}{BB'} = \ucobmtrx{B}{B'} \matrixOf{T}{B} \ucobmtrx{B'}{B} = \uinvcobmtrx{B'}{B} \matrixOf{T}{B} \ucobmtrx{B'}{B}\text{.}\label{equation-lintrans-similarity-cob}\tag{\(\star\)} \end{gather}

In other words, matrices of an operator relative to different bases are similar. Restricting our attention to \(V = \R^n\) (or \(V = \C^n\) in the complex case), we can turn this around and say that similar matrices represent the same linear operator relative to different bases, and a similarity relation just realizes a change of basis between the two representations of the operator.

Finally, note that the change-of-basis perspective in (\(\star\)) agrees with the calculation pattern

\begin{equation*} \matrixOf{T(\uvec{v})}{C} = \matrixOf{T}{C} \matrixOf{\uvec{v}}{C} \end{equation*}

that holds for every basis \(\basisfont{C}\) of the domain space \(V\text{.}\) Indeed, for every vector \(\uvec{v}\) in \(V\text{,}\) we have

\begin{align*} \matrixOf{T}{B'} \matrixOf{\uvec{v}}{B'} \amp = \ucobmtrx{B}{B'} \matrixOf{T}{B} \ucobmtrx{B'}{B} \matrixOf{\uvec{v}}{B'} \\ \amp = \ucobmtrx{B}{B'} \matrixOf{T}{B} \matrixOf{\uvec{v}}{B} \\ \amp = \ucobmtrx{B}{B'} \matrixOf{T(\uvec{v})}{B} \\ \amp = \matrixOf{T(\uvec{v})}{B'} \text{,} \end{align*}

as required.

Subsection 46.2.2 Transferring matrix concepts to operators

We have seen that similar matrices are essentially the same matrix, just with different entries — similar matrices have the same determinant, trace, rank, nullity, characterisic polynomial, and eigenvalues, and their column spaces, null spaces, and eigenspaces can be related through a suitable transition matrix. This allows us to unambiguously transfer or relate these concepts to corresponding concepts for operators.

For example, we can define the determinant of an operator \(T\) on a finite-dimensional space \(V\) to be the determinant of the matrix \(\matrixOf{T}{B}\) relative to a basis \(\basisfont{B}\) for \(V\text{.}\) This definition is unambiguous even though it seems to depend on what basis for \(V\) is chosen, because matrices for an operator relative to different bases are always similar, so those different matrices will always have the same determinant as each other. And similarly for the trace of an operator.

For eigenvalues and eigenvectors, we can make initial definitions based on the same pattern as for matrices instead of using matrices to make the definition. So an eigenvalue/eigenvector pair for a linear operator \(T\) on a vector space \(V\) (and here, for the purposes of the definition, we do not need \(V\) to be finite-dimensional) is a scalar \(\lambda\) and a nonzero vector \(\uvec{v}\) in \(V\) so that

\begin{equation*} T(\uvec{v}) = \lambda \uvec{v} \text{.} \end{equation*}

Manipulating this definition in the same way as for matrices, we see that we have an eigenvalue/eigenvector pair precisely when

\begin{equation*} (\lambda I_V - T)(\uvec{v}) = \zerovec_V \text{,} \end{equation*}

which leads to the familiar definition of the characterisic polynomial of an operator as

\begin{equation*} \det (\lambda I_V - T) \text{,} \end{equation*}

which is now meaningful based on the idea of a determinant of an operator. And it will be straightforward to connect these new concepts for operators to the corresponding concepts for the matrix \(\matrixOf{T}{B}\) for arbitrary choice of basis \(\basisfont{B}\) of \(V\text{.}\)

For rank and nullity, the connection is direct. We already have these concepts for both matrices and general linear transformations (not just operators), and we know they coincide when we create a matrix out of a transformation (Proposition 45.5.4).

Subsection 46.2.3 Canonical forms of linear operators

Taking our analogies between linear operators and their matrices even further, using similarity of operators via change of basis we can identify

  • diagonalizable operators as those that have a matrix representation relative to some basis that a diagonal matrix;
  • block-diagonal form for operators relative to a complete set of independent, operator-invariant subspaces of the domain space;
  • generalized eigenvectors and eigenspaces of an operator that correspond to a matrix for the operator in triangular-block form;
  • nilpotent operators and collections of cyclic subspaces of the domain space that correspond to a matrix for the operator that is in elementary nilpotent form or has blocks of elementary nilpotent submatrices; and
  • Jordan form for operators corresponding to a choice of basis for which the matrix of the operator is in Jordan normal form.