Section 29.1 Motivation
We continue our quest to answer Question 28.1.1.
In Chapter 28, we branched off from diagonal form to a generalization: block-diagonal form. But if we were to ask what the simplest form of matrix after diagonal form is, the answer that likely comes to mind is some kind of triangular form, upper or lower. Since there is no difference in the relative simplicity of the two, we will pursue upper triangular forms for now (though we will have reason to switch to lower triangular forms later).
Philosophy of inquiry: an incrementally more difficult case.
Take the patterns we are already familiar with and try something slightly more complicated, but maybe aim for the simplest form of “slightly more complicated.”
Triangular form will definitely be more complicated than diagonal form. What kind of a triangular form will be only slightly more complicated? We have seen that eigenvalues and eigenvectors play an important role in diagonal form, and perhaps they will be needed again for triangular form. Computing eigenspaces for different eigenvalues is tedious, and it's not clear how the different eigenspaces will fit together to create a triangular form. Let's simplify out initial study of triangular form by only considering matrices that have a single eigenvalue. If such a matrix is similar to a triangular form matrix, the triangular matrix will also have only a single eigenvalue (Theorem 26.5.8). Since the eigenvalues of a triangular matrix are precisely the diagonal entries, the triangular form matrix must have that eigenvalue repeated down the diagonal. Here's an example of a \(3 \times 3\) upper triangular matrix of this form:
We will refer to this form of matrix as scalar-triangular.