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Matrix Algebra

Learn the basics of matrix algebra including addition, multiplication, transposition, inverses, determinants, and applications in solving linear equations. Understand how matrices represent data and perform operations. Explore Cramer’s rule for solving systems of equations.

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Matrix Algebra

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  1. Matrix Algebra Methods for Dummies FIL November 17 2004 Mikkel Wallentin mikkel@pet.auh.dk

  2. Sources • www.sosmath.com • www.mathworld.wolfram.com • www.wikipedia.org • Maria Fernandez’ slides (thanks!) from previous MFD course: http://www.fil.ion.ucl.ac.uk/spm/doc/mfd-2004.html • Slides from SPM courses:http://www.fil.ion.ucl.ac.uk/spm/course/

  3. Design matrix … = the betas (here : 1 to 9) parameters error vector design matrix data vector a m b3 b4 b5 b6 b7 b8 b9 = + ´ = + Y X b e

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

  5. 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.

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

  7. Multiplication of a matrix and a constant: 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.

  8. …Each parameter (the betas) assigns a weight to a single column in the design matrix … = the betas (here : 1 to 9) parameters error vector design matrix data vector a m b3 b4 b5 b6 b7 b8 b9 = + ´ = + Y X b e

  9. column → row row →column Transposition

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

  11. …A contrast estimate is obtained by multiplying the parameter estimates by a transposed contrast vector … parameters error vector design matrix contrast vector data vector 1 0 0 0 0 0 0 0 0 a m b3 b4 b5 b6 b7 b8 b9 = + c + e ´ = Y X b

  12. cT=1 0 0 0 0 0 0 0 contrast ofestimatedparameters cTb T = T = varianceestimate s2cT(XTX)+c SPM{t} T test - one dimensional contrasts - SPM{t} A contrast= a linear combination ofparameters: cT´b box-car amplitude> 0 ? = b1> 0 ? => b1b2b3b4b5.... Compute 1xb1+0xb2+0xb3+0xb4+0xb5+ . . . and divide by estimated standard deviation

  13. 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 matrixA, we haveA In = InA = A • For any nxm matrix A, we have InA = A, and A Im = A

  14. H0: True model is X0 X0 X0 X1(b3-9) 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 This model ? Or this one ? F-test (SPM{F}) : a reduced model or ...multi-dimensional contrasts ? tests multiple linear hypotheses. Ex : does DCT set model anything? test H0 : cT´ b = 0 ? H0: b3-9 = (0 0 0 0 ...) cT = SPM{F}

  15. 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

  16. Determinants • Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations (i.e. GLMs). • The determinant is a function that associates a scalar det(A) to every square matrixA. • The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear 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: Recall that for 2x2 matrices: And generally :

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

  18. 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

  19. A similar problem … = the betas (here : 1 to 9) parameters error vector design matrix data vector a m b3 b4 b5 b6 b7 b8 b9 = + ´ = + Y X b e

  20. 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!

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