Maths4ml linearalgebra-formula
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This document provides a summary of key linear algebra concepts for machine learning, including: 1) Vector operations such as addition, scalar multiplication, dot products, and projections. 2) Matrix operations including matrix multiplication, identity matrices, determinants, and inverses. 3) Basis definitions and the Gram-Schmidt process for constructing orthonormal bases. 4) Eigenvalue problems and how they relate to characteristics of matrices like finding the PageRank vector through the power method.
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Maths4ml linearalgebra-formula
- 1. Mathematics for Machine Learning: Linear Algebra Formula Sheet Vector operations r + s = s + r 2r = r + r r 2 = i r2 i - dot or inner product: r.s = i risi commutative r.s = s.r distributive r.(s + t) = r.s + r.t associative r.(as) = a(r.s) r.r = r 2 r.s = r s cos θ - scalar and vector projection: scalar projection: r.s r vector projection: r.s r.r r Basis A basis is a set of n vectors that: (i) are not linear combinations of each other (ii) span the space The space is then n-dimensional. Matrices Ar = r a b c d e f = ae + bf ce + df A(nr) = n(Ar) = nr A(r + s) = Ar + As Identity: I = 1 0 0 1 clockwise rotation by θ: cos θ sin θ − sin θ cos θ determinant of 2x2 matrix: det a b c d = ad − bc inverse of 2x2 matrix: a b c d −1 = 1 ad − bc d −b −c a - summation convention for multiplying matrices a and b: abik = j aijbjk Change of basis Change from an original basis to a new, primed basis. The columns of the transformation matrix B are the new basis vectors in the original coordinate system. So Br = r where r is the vector in the B-basis, and r is the vector in the original basis. Or; r = B−1 r If a matrix A is orthonormal (all the columns are of unit size and orthogonal to eachother) then: AT = A−1 Gram-Schmidt process for constructing an orthonormal basis Start with n linearly independent basis vectors v = {v1, v2, ..., vn}. Then e1 = v1 ||v1|| u2 = v2 − (v2.e1)e1 so e2 = u2 ||u2|| ... and so on for u3 being the remnant part of v3 not composed of the preceding e-vectors, etc. ... Transformation in a Plane or other object First transform into the basis referred to the reflection plane, or whichever; E−1 . Then do the reflection or other transformation, in the plane of the object TE. Then transform back into the original basis E. So our transformed vector r = ETEE−1 r. Eigenstuff To investigate the characteristics of the n by n matrix A, you are looking for solutions the the equation, Ax = λx where λ is a scalar eigenvalue. Eigenvalues will satisfy the following condition (A − λI)x = 0 where I is an n by n dimensional identity matrix - PageRank To find the dominant eigenvector of link matrix L, the Power Method can be iteratively applied, starting from a uniform initial vector r. ri+1 = Lri A damping factor, d, can be implement to stabilize this method as follows. ri+1 = dLri + 1 − d n