This is a little-known gem among linear algebra texts. I recommend it to professionals all the time. Here is why: Most practitioners have a basic knowledge of matrices and linear algebra. They  know Gauss-Jordan elimination and understand what a singular matrix is. They want a book that reviews the basics, but opens doors to more advanced topics that elementary books touch only lightly or not at all. This book is perfect. Its title says it all: Matrix Theory: A Second Course. Its first chapter reviews concepts covered in any introductory linear algebra course: matrices. vectors, determinants, linear systems, inverses, eigenvalues, and eigenvectors. If you have seen these before and only need to brush up, the chapter is perfect. Chapter 2 covers linear spaces and operators. Euclidean space is a familiar linear space. This chapter generalizes the notion. Chapter 3 covers canonical forms. It addresses the question of how to change a coordinate system in order to simplify a matrix—say to diagonalize the matrix, as is done to covariance matrices by principal component analysis. Chapter 4 covers quadratic polynomials and optimization, mathematics that underlies the method of least squares. Chapter 5 covers difference and differential equations. The last chapter covers a variety of special topics, including nonnegative matrices, higher-order eigenvalue problems, matrix equations and special matrices, including positive-definite matrices.

The book covers a lot of sophisticated material, but it is very practical. Abstract concepts are presented only if they add insight. Discussions are rooted in matrix theory. If you are looking for an advanced book that downplays matrices, and emphasized more abstract theory leading towards functional analysis, consider Lad (1996) instead. For applications-oriented professionals—for self study or as a reference—Ortega is the book to have.