This chapter introduces and methodically develops the important and useful topic of bidiagonal factorization. Factorization of matrices is one of the most important topics in matrix theory, and plays a central role in many related applied areas such as numerical analysis and statistics. Investigating when a class of matrices admits a particular type of factorization is an important study, which historically has been fruitful. Often many intrinsic properties of a particular class of matrices can be deduced via certain factorization results. For example, it is a well-known fact that any (invertible) M-matrix can be factored into a product of a lower triangular (invertible) M-matrix and an upper triangular (invertible) M-matrix. This LU factorization result leads to the conclusion that the class of M-matrices is closed under Schur complementation, because of the connection between LU factorizations and Schur complements. This chapter focuses on triangular factorization extended beyond just LU factorization, however.
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