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Chapter 1 Research History of Sign Patterns That Allow Diagonalizability 1.1 Introduction 1.2 Preliminaries Chapter 2 Sign Patterns That Allow Diagonalizability 2.1 Normal sign patterns 2.2 Necessary and sufficient conditions for allowing diagonalizability Chapter 3 Rank Conditions for Sign Patterns That Allow Diagonalizability 3.1 Introduction and preliminaries 3.2 Main results Chapter 4 The Sign K-potent Sign Pattern Matrix That Allows Diagonalizability 4.1 Necessary and sufficient conditions of allowing diagonalizability 4.2 Sign idempotent sign pattern matrix 4.3 Sign pattern matrices that are power Chapter 5 New Results on Sign Patterns That Allow Diagonalizability 5.1 Introduction and preliminaries 5.2 Necessary and/or sufficient conditions for allowing diagonalizability 5.3 Entry modifications between matrices achieving extreme ranks 5.4 Further results on sign patterns that allow diagonalizability 5.5 Sign patterns in Frobenius normal form Chapter 6 Some Attempts of Sign Patterns That Allow Diagonalizability from Different Direction 6.1 Sign patterns allowing simultaneous unitary diagonalizability 6.2 Up to 4 order sign patterns that allow diagonalizability 6.3 Recent advance on sign patterns with special graphs that allow diagonalizability 6.4 Research direction of sign patterns that allows diagonalizability in the future Notation Index References Acknowledge