Matrix Algebra Methods for Dummies FIL November Mikkel Wallentin

Sources Maria Fernandez’ slides (thanks!) from previous MFD course: html html Slides from SPM courses:

Design matrix … =  +  =  + YX data vector design matrix parameters error vector  = the betas (here : 1 to 9)

Scalars, vectors and matrices Scalar: Variable described by a single number – e.g. Image intensity (pixel value) Vector: Variable described by magnitude and direction Square (3 x 3)Rectangular (3 x 2) d i j : i th row, j th column 3 2 Matrix: Rectangular array of scalars

Matrices A matrix is defined by the number of Rows and the number of Columns (eg. a (mxn) matrix has m rows and n columns). A square matrix of order n, is a (nxn) matrix.

Addition (matrix of same size) –Commutative: A+B=B+A –Associative: (A+B)+C=A+(B+C) Eg. Matrix addition

Matrix multiplication Rule: In order to perform the multiplication AB, where A is a (mxn) matrix and B a (kxl) matrix, then we must have n=k. The result will be a (mxl) matrix. Multiplication of a matrix and a constant:

…Each parameter (the betas) assigns a weight to a single column in the design matrix … =  +  =  + YX data vector design matrix parameters error vector  = the betas (here : 1 to 9)

Transposition column → rowrow → column

Outer product = matrix Inner product = scalar Two vectors: Example Note: (1xn)(nx1) -> (1X1) Note: (nx1)(1xn) -> (nXn)

…A contrast estimate is obtained by multiplying the parameter estimates by a transposed contrast vector … =  +  =  + YX data vector design matrix parameters error vector   contrast vector c

SPM{t} A contrast = a linear combination of parameters: c T   c T = divide by estimated standard deviation T test - one dimensional contrasts - SPM{t} T = contrast of estimated parameters variance estimate T = s 2 c(XX) + c s 2 c T (X T X) + c cbcTbcbcTb box-car amplitude > 0 ? =   > 0 ? => Compute 1 x b  + 0 x b  + 0 x b  + 0 x b  + 0 x b  +... and b  b  b  b  b ....

Identity matrices Is there a matrix which plays a similar role as the number 1 in number multiplication? Consider the nxn matrix : For any nxn matrix A, we have A I n = I n A = A For any nxm matrix A, we have I n A = A, and A I m = A

H 0 :  3-9 = ( ) c T = SPM{F} tests multiple linear hypotheses. Ex : does DCT set model anything? F-test (SPM{F}) : a reduced model or... multi-dimensional contrasts ? test H 0 : c T  b = 0 ? X 1 (  3-9 ) X0X0 This model ?Or this one ? H 0 : True model is X 0 X0X

Inverse matrices Definition. A matrix A is called nonsingular or invertible if there exists a matrix B such that: Notation. A common notation for the inverse of a matrix A is A -1. So: The inverse matrix is unique when it exists. So if A is invertible, then A -1 is also invertible and

Determinants Recall that for 2x2 matrices: Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations (i.e. GLMs).systems of linear equations The determinant is a function that associates a scalar det(A) to every square matrix A.functionscalarsquare matrix The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation.volumelinear transformation A matrix A has an inverse matrix A -1 if and only if det(A)≠0. Determinants can only be found for square matrices. For a 2x2 matrix A, det(A) = ad-bc. Lets have at closer look at that: And generally :

Matrix Inverse - Calculations A general matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination or LU decomposition i.e. Note: det(A)≠0

System of linear equations Imagine a drink made of egg, milk and orange juice. Some of the properties of these ingredients are described in this table: If we now want to make a drink with 540 calories and 25 g of protein, the problem of finding the right amount of the ingredients can be formulated like this: or

A similar problem … =  +  =  + YX data vector design matrix parameters error vector  = the betas (here : 1 to 9)

Cramer’s rule Consider the linear system (in matrix form) A X = B where A is the matrix coefficient, B the nonhomogeneous term, and X the unknown column-matrix. We have: Theorem. The linear system AX = B has a unique solution if and only if A is invertible. In this case, the solution is given by the so-called Cramer's formulas: where xi are the unknowns of the system or the entries of X, and the matrix Ai is obtained from A by replacing the ith column by the column B. In other words, we have where the bi are the entries of B. Thank you Bent Kramer!