11: Canonical Forms — Read — Invisible Network

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The work in Chapter 9 is continued. Invariant subspaces and direct sums are used to derive the block triangular form. That, in turn, is used to give a compact proof of the Jordan canonical form. Of course the level is higher.

11.0: Prelude to Canonical Forms
11.1: Block Triangular Form
This page covers the Block Triangulation Theorem, showing how an \( n \times n \) matrix with real eigenvalues can be transformed into a block upper triangular form using an invertible matrix. It explains the construction of bases for generalized eigenspaces, outlines a triangulation algorithm, and discusses the Cayley-Hamilton theorem.
- 11.1E: Block Triangular Form Exercises
11.2: The Jordan Canonical Form
This page covers row-equivalence and canonical forms of matrices, emphasizing the unique reduced row-echelon form and Smith normal form through row and column operations. It explains the Jordan canonical form, helping to represent matrices with real eigenvalues as block triangular matrices using Jordan blocks for nilpotent operators.
- 11.2E: The Jordan Canonical Form Exercises

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