Chapter 7: EigenStuff

Eigen (pronounced EYE-GUN) is a German prefix to words such as eigentum (property), eigenschaft (a feature or characteristic), and eigensinn (an idiosyncrasy). To sum up, we are looking for some values and vectors that are characteristic, idiosyncratic, and a property of something. What is that something? That something is our old friend the linear operator and its representations as square matrices. But before we get there, we'll need to look at some other concepts such as the matrix inverse and determinant. We'll wrap it all up with the trace of a matrix and some properties that the trace, determinant, and eigenvalues all share. These concepts will allow us to reach even further heights in the chapters that follow.

In this chapter, we are going to cover the following main topics:

• The inverse of a matrix
• Determinants
• Invertible matrix theorem
• Eigenvalues and eigenvectors
• Trace
• The special properties of eigenvalues

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