1. Matrix Algebra (and why it’s important!)
Methods for Dummies
FIL
October 2007
Steve Fleming & Verity Leeson
Need to add: division, and why its important. Graphical model of determinant. Animate slide 4.

2. Sources and further information
Jon Machtynger & Jen Marchant’s slides!
Human Brain Function textbook (for GLM)
SPM course
Web Guides
(Formal Modelling in Cognitive Science course)

3. 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
Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. A vector, here, is a directed line segment, characterized by both its magnitude, represented by length, and its direction. Vectors can be used to represent physical entities such as forces, and they can be added to each other and multiplied with scalars, thus forming the first example of a real vector space.
Modern linear algebra has been extended to consider spaces of arbitrary or infinite dimension. Four dimensional vector is image at time point in 3d space
Matrix: Rectangular array of vectors defined by number of rows and columns
Square (3 x 3) Rectangular (3 x 2) d r c : rth row, cth column 3 2
(Roman Catholic)

4. Matrices in Matlab Vector formation: [1 2 3] Matrix formation:
‘;’ is used to signal end of a row
‘:’ is used to signify all rows or columns
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…
Magic is where rows, columns and diagonals all sum to same total. Has to be square.

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
a11
a12
a13
b11
b12
a21
a22
a23 X
b21
b22
a31
a32
a33
b31
b32
a41
a42
a43 m k
Matrix multiplication rule:
“When A is a mxn matrix & B is a kxl matrix, AB is only viable if n=k. The result will be an mxl matrix”
b11
b12
a11
a12
a13
b21
b22 x
a21
a22
a23
b31
b32
a31
a32
a33
a41
a42
a43

7. Multiplication method
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:
Inner product = scalar
(1xn)(nx1)  (1X1)
Outer product = matrix
(nx1)(1xn)  (nXn)

10. Identity matrices
Is there a matrix which plays a similar role as the number 1 in number multiplication?
Consider the nxn matrix:
A square nxn matrix A has one
A In = In A = A
An nxm matrix A has two!!
In A = A & A Im = A
Worked example
A In = A
for a 3x3 matrix: 1 2 3
1+0+0
0+2+0
0+0+3 4 5 6 X =
4+0+0
0+5+0
0+0+6 7 8 9
7+0+0
0+8+0
0+0+9
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: 1 X
2 3
-1 3 = -1 2
1 3
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: In Matlab: det(A) = det(A)
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)!
thus
Note: det(A)≠0
More complex matrices can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination or LU decomposition

14. Applications

15. SEM
Neural Networks
SPM

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

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. Matrix Representations of Neural Connections –Scenario 2 again#1 #3 #2 + = -1 +1 1
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. Input vector = (1 1) relates to activity (#1 #2) #3 #2 + = -1 +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. Y = X . β + ε Y is a matrix of BOLD signals
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
Time
Intensity

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

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

30. The End…
Any (easy) questions?!