Appendix A: Mathematical Foundations

Mathematical foundations O ur goal here is to present the basic results and definitions from linear algebra, probability theory, information theory and computational complexity that serve as the mathematical foundations for pattern recognition. We will try to give intuitive insight whenever appropriate, but do not attempt to prove these results; systematic expositions can be found in the references. Here are the terms and notation used throughout the book. In addition, there are numerous specialized variables and functions whose definitions and usage should be clear from the text. variables, symbols and operations approximately equal to ≡ equivalent to (or defined to be) ∝ proportional to ∞ infinity x → a xapproaches a t ← t + 1 in an algorithm: assign to variable t the new value t + 1 lim x→a f (x) the value of f (x) in the limit as x approaches a arg max x f (x) the value of x that leads to the maximum value of f (x) arg min x f (x) the value of x that leads to the minimum value of f (x) x ceiling of x, i.e., the least integer not smaller than x (e.g., 3.5 = 4) x floor of x, i.e., the greatest integer not larger than x (e.g., 3.5 = 3) m mod n mmodulo n, the remainder when m is divided by n (e.g., 7 mod 5 = 2) ln(x) logarithm base e, or natural logarithm of x log(x) logarithm base 10 of x log 2 (x) logarithm base 2 of x 3