Linear Algebra, Principal Component Analysis and their Chemometrics Applications

Linear algebra is the language of chemometrics. One can not expect to truly understand most chemometric techniques without a basic understanding of linear algebra Linear Algebra

Vector A vector is a mathematical quantity that is completely described by its magnitude and direction x1x1 y1y1 x y P

Vector A vector is a mathematical quantity that is completely described by its magnitude and direction x1x1 y1y1 x y P P = x1x1 y1y1

MATLAB Notation

Length of a Vector x1x1 y1y1 x y P P = x y 1 2 x = [x 1, x 2, …, x n ] x M = ( x i 2 ) 0.5 i=1 n Normal Vector u = x x u =1

Normalized vector

Mean Centered Vector x1x1 x2x2 xnxn … x = mx = M xixi i=1 n n mcx = x 1 - mx … x 2 - mx x n - mx x = mx = mcx = y = = mx = 3

Mean centered

? The length of a mean centered vector is proportional to the standard deviation of its elements y1y1 y2y2 ynyn … y =y* = y 1 - m y … y 2 - m y y n - m y y* ≈ s y i

A set of p vectors [x 1, x 2, …, x p ] with same dimension n is linearly independent if the expression: c i x i = 0 M i=1 p holds only when all p coefficients c i are zero Linear Independent Vectors x1x1 x2x2 x3x3 c1x1c1x1 c2x2c2x2

A vector space spanned by a set of p linearly independent vectors (x 1, x 2, …, x p ) with the same dimension n is the set of all vectors that are linear combinations of the p vectors that span the space Vector Space Basis set A set of n vectors of dimension n which are linearly independent is called a basis of an n-dimensional vector space. There can be several bases of the same vector space A coordinate space can be thought of as being constructed from n basis vectors of unit length which originate from a common point and which are mutually perpendicular Coordinate Space

Q =  P =  x 1  y 1  x 1  y 1 x y P x1x1 y1y1 Q Vector Multiplication by a Scalar

x = 1.19 y = 2.38 x y y = 2 x

Addition of Vectors x1x1 x2x2 xnxn … y1y1 y2y2 ynyn … x + y = + = x 1 + y 1 … x 2 + y 2 x n + y n x + y x1x1 x2x2 y1y1 y2y2 y x x 1 + y 1 x 2 + y 2

 x1 c x … a x + a y = + =  x2 c x  xn c x  y1 c y …  y2 c y  yn c y  x1 c x +  y1 c y …  x2 c x +  y2 c y  xn c x +  yn c y Component 1Component 2mixture axax ayay a x + a y

Subtraction of Vectors x1x1 x2x2 xnxn … y1y1 y2y2 ynyn … x - y = - = x 1 - y 1 … x 2 - y 2 x n - y n x - y x1x1 x2x2 y1y1 y2y2 y x

Inner Product (Dot Product) x1x1 x2x2 xnxn … x. x = x T x =[x 1 x 2 … x n ]= x x … +x n 2 = x 2 x. y = x T y = x y cos  The cosine of the angle of two vectors is equal to the dot product between the normalized vectors: x. y x y cos  =

y x x. y = x y y x x. y = - x y y x x. y = 0 Two vectors x and y are orthogonal when their scalar product is zero x. y = 0and xy = 1= Two vectors x and y are orthonormal