In this chapter we recall some needed mathematical background materials. These include the fundamental facts and properties of vector spaces and matrix theory and the definitions and properties of various norms of vectors, matrices, signals and rational transfer functions. These materials are particularly useful for developing and understanding results in the following chapters. More detailed information on the subjects can be found in more specific mathematics textbooks, or other texts in linear systems and control theory (see, e.g., Barnett [8], Chen [22], Chen [33], Desoer and Vidyasagar [49], Golub and Van Loan [59], Huang [68], Kailath [70], Kreyszig [78], Saberi et al. [120], Suda et al. [138], Trentelman et al. [141], Wielandt [150] and Wonham [154]).