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Understanding Matrix Algebra: Essential Concepts and Applications

This guide explores the fundamentals of matrix algebra, highlighting its critical role in various fields, including cognitive science and neural networks. Learn about scalars, vectors, matrices, and operations such as addition, subtraction, multiplication, and finding inverses. Discover how matrices model complex systems, including neural connections, and the importance of determinants in solving equations. Resources for further study, including online courses and textbooks, are also provided to enhance your understanding of this essential mathematical tool.

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Understanding Matrix Algebra: Essential Concepts and Applications

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  1. Matrix Algebra (and why it’s important!) Methods for Dummies FIL October 2007 Steve Fleming & Verity Leeson

  2. Sources and further information • Jon Machtynger & Jen Marchant’s slides! • Human Brain Function textbook (for GLM) • SPM course http://www.fil.ion.ucl.ac.uk/spm/course/ • Web Guides • http://mathworld.wolfram.com/LinearAlgebra.html • http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html • http://www.inf.ed.ac.uk/teaching/courses/fmcs1/ (Formal Modelling in Cognitive Science course) • http://www.wikipedia.org

  3. 2 3 (Roman Catholic) Square (3 x 3) Rectangular (3 x 2) d r c : rthrow, cthcolumn Scalars, vectors and matrices • Scalar:Variable described by a single number – e.g. Image intensity (pixel value) • Vector: Variable described by magnitude and direction – e.g. Image intensity at a particular time • Matrix: Rectangular array of vectors defined by number of rows and columns

  4. Matrices in Matlab ‘;’ is used to signal end of a row ‘:’ is used to signify all rows or columns • Vector formation: [1 2 3] • Matrix formation: X = [1 2 3; 4 5 6; 7 8 9] = Subscripting – each element of a matrix can be addressed with a pair of numbers; row first, column second (Roman Catholic) e.g. X(2,3) =6 X(3, :) = X( [2 3], 2) = • “Special” matrix commands: • zeros(3,1) = • ones(2) = • magic(3) • more to come…

  5. Matrix addition Addition (matrix of same size) • Commutative: A+B=B+A • Associative: (A+B)+C=A+(B+C) Subtraction (consider as the addition of a negative matrix)

  6. Matrix multiplication • Scalar multiplication: • Rule for multiplication of vectors/matrices: n l m k Matrix multiplication rule: “When A is a mxnmatrix & B is a kxl matrix, AB is only viable if n=k. The result will be an mxl matrix”

  7. Multiplication method l l • Sum over product of respective rows and columns • For larger matrices, following method might be helpful: m m Define output matrix X = r c Sum over crc = • Matlab does all this for you! • Simply type: C = A * B • N.B. If you want to do element-wise multiplication, use: A .* B =

  8. Transposition column → row row →column Mrc = Mcr • In Matlab: AT = A’

  9. Outer and inner products of vectors Two vectors: (1xn)(nx1)  (1X1) Inner product = scalar Outer product = matrix (nx1)(1xn)  (nXn)

  10. Worked example A In = A for a 3x3 matrix: Identity matrices • Is there a matrix which plays a similar role as the number 1 in number multiplication? Consider the nxnmatrix: • A square nxn matrix Ahas one • A In = InA = A • An nxm matrix A has two!! • InA = A & A Im = A • In Matlab:eye(r, c) produces an r x c identity matrix

  11. Inverse matrices • Definition. A matrix A is nonsingular or invertible if there exists a matrix B such that: worked example: • Common notation for the inverse of a matrix A is A-1 • The inverse matrix A-1 is unique when it exists. • If A is invertible, A-1 is also invertible  A is the inverse matrix of A-1. • Matrix division: • A/B = AB-1 • If A is an invertible matrix, then (AT)-1 = (A-1)T • In Matlab:A-1 = inv(A)

  12. Determinants • Determinant is a function: • Input is nxn matrix • Output is a single number (real or complex) called the determinant • A matrix A has an inverse matrix A-1 if and only if det(A)≠0 (see next slide) • In Matlab:det(A) = det(A)

  13. Calculation of inversion using determinants Or you can just type inv(A)! Note: det(A)≠0 thus More complex matrices can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination or LU decomposition

  14. Applications

  15. SEM http://www.maths.soton.ac.uk/~jav/soton/MATH1007/workbook_8/8_2_inv_mtrx_sim_lin_eqnpdf.pdf • Neural Networks http://csn.beckman.uiuc.edu/k12/nn_matrix.pdf • SPM http://imaging.mrc-cbu.cam.ac.uk/imaging/PrinciplesStatistics

  16. Solving simultaneous equations • For one linear equation ax=b where the unknown is x and a and b are constants • 3 possibilities

  17. With >1 equation and >1 unknown • Can use solution from the single equation to solve • For example • In matrix form AX = B

  18. Need to find determinant of matrix A (because X =A-1B) • From earlier • (2 x -2) – (3 x 1) = -4 – 3 = -7 • So determinant is -7 • To find A-1

  19. if B is So

  20. Excitatory Connection Inhibitory Connection Input Neuron Output Neuron Input Neuron Output Neuron Neural Networks • Neural networks create a mathematical model of the connections in a neural system • Connections are the excitatory and inhibitory synapses between neurons

  21. Scenario 1 Input Neuron Output Neuron then If then If

  22. + = Scenario 2 • The combination of both an active excitatory and active inhibitory input will cancel out • No net activity

  23. #2 -1 +1 #1 #3 1 + 1 = Matrix Representations of Neural Connections –Scenario 2 again Excitatory = positive influence on post synaptic cell Inhibitory = negative influence With the synapses labelled (1-3) and activity level specified we can translate this information into a set of vectors (1 row matrices)

  24. #2 -1 +1 #1 #3 1 + 1 = • Input vector = (1 1) relates to activity (#1 #2) • Weight vector = (1 -1) relates to connection weight (#1 #2) Activity of Neuron 3 Input x weight With varying input (activity) and weight, neuron 3 can take on a wide range of values

  25. How are matrices relevant to fMRI data? • Consider that data measured includes • Response variable e.g BOLD signal at a particular voxel Many scalars for this one voxel • Explanatory variables These are assumed to be measured without error May be continuous May be dummy indicating levels of an experimental factor

  26. With a single explanatory variable Y= X . β+ ε Observed = Predictors * Parameters + Error BOLD = Design Matrix * Betas + Error

  27. Time Intensity Y = X . β + ε Preprocessing ... Y • Y is a matrix of BOLD signals • Each column represents a single voxel sampled at successive time points. • Each voxel is considered as independent observation • Analysis of individual voxels over time, not groups over space

  28. Design Matrix Y = X . β + ε Y X1 X2 X1 X2 Most –ve nearest black, most +ve nearest white Matrix Rows : values of X for a single predictor Columns : different predictors

  29. Solve equation for β – tells us how much of the BOLD signal is explained by X A complex version data vector (Voxel) parameters design matrix error vector a m b3 b4 b5 b6 b7 b8 b9 = + × = + Y X b e

  30. The End… Any (easy) questions?!

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