![Matrices
• A matrix is defined by the number of Rows and the
number of Columns.
• An mxn matrix has m rows and n columns.
A = 4x3 matrix
• A square matrix of order n, is an nxn matrix.
21 2 53
5 34 12
6 33 55
74 27 3
Matlab notes ( ; End of matrix row )
A = [ 21 5 53 ; 5 34 12 ; 6 33 55 ; 74 27 3 ]
To extract data: Matrix name( row, column )
Scalar Data Point A( 1 , 2 ) = 2
Row Vector A( 2 , : ) = [ 5 34 12 ]
Column Vector A( : , 3 ) = [ 53 ; 12 ; 55 ; 3 ]
Smaller Matrix A(2:4,1:2) = [ 5 34 ; 6 33 ; 74 27 ]
Another Matrix A( 2:2:4 , 2:3 ) = [ 34 12 ; 27 3 ]](https://image.slidesharecdn.com/linearalgebramatrices-240621122226-1b991638/85/Linear-Algebra-and-Matrices-Powerpoint-2024-4-320.jpg)
Linear Algebra and Matrices Powerpoint 2024
- 1. Matrix Algebra Methods for Dummies FIL January 25 2006 Jon Machtynger & Jen Marchant
- 2. Acknowledgements / Info • Mikkel Walletin’s (Excellent) slides • John Ashburner (GLM context) • Slides from SPM courses: http://www.fil.ion.ucl.ac.uk/spm/course/ • Good Web Guides – www.sosmath.com – http://mathworld.wolfram.com/LinearAlgebra.html – http://ceee.rice.edu/Books/LA/contents.html – http://archives.math.utk.edu/topics/linearAlgebra.html
- 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 Square (3 x 3) Rectangular (3 x 2) d r c : rth row, cth column 3 2 • Matrix: Rectangular array of scalars
- 4. Matrices • A matrix is defined by the number of Rows and the number of Columns. • An mxn matrix has m rows and n columns. A = 4x3 matrix • A square matrix of order n, is an nxn matrix. 21 2 53 5 34 12 6 33 55 74 27 3 Matlab notes ( ; End of matrix row ) A = [ 21 5 53 ; 5 34 12 ; 6 33 55 ; 74 27 3 ] To extract data: Matrix name( row, column ) Scalar Data Point A( 1 , 2 ) = 2 Row Vector A( 2 , : ) = [ 5 34 12 ] Column Vector A( : , 3 ) = [ 53 ; 12 ; 55 ; 3 ] Smaller Matrix A(2:4,1:2) = [ 5 34 ; 6 33 ; 74 27 ] Another Matrix A( 2:2:4 , 2:3 ) = [ 34 12 ; 27 3 ]
- 5. 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 Matrix addition
- 6. Matrix multiplication Matrix multiplication rule: When A is a mxn matrix & B is a kxl matrix, the multiplication of AB is only viable if n=k. The result will be an mxl matrix. Constant (or Scalar) multiplication of a matrix:
- 7. Visualising multiplying b11 b12 b21 b22 b31 b32 a11 a12 a13 a11b11 + a12b21 + a13b31 a11b12 + a12b22 + a13b32 a21 a22 a23 a21b11 + a22b21 + a23b31 a21b12 + a22b22 + a23b32 a31 a32 a33 a31b11 + a32b21 + a33b31 a31b12 + a32b22 + a33b32 a41 a42 a43 a41b11 + a42b21 + a43b31 a41b12 + a42b22 + a43b32 a11 a12 a13 b11 b12 ? ? a21 a22 a23 X b21 b22 = ? ? a31 a32 a33 b31 b32 ? ? a41 a42 a43 ? ? A matrix = ( m x n ) B matrix = ( k x l ) A x B is only viable if k = n width of A = height of B Result Matrix = ( m x l ) Jen’s way of visualising the multiplication
- 8. Transposition column → row row → column Mrc = Mcr
- 9. Outer product = matrix Inner product = scalar Two vectors: Example Note: (1xn)(nx1) (1X1) Note: (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 1 2 3 1 0 0 1+0+0 0+2+0 0+0+3 4 5 6 X 0 1 0 = 4+0+0 0+5+0 0+0+6 7 8 9 0 0 1 7+0+0 0+8+0 0+0+9 Worked example A In = A for a 3x3 matrix:
- 11. Inverse matrices • Definition. A matrix A is nonsingular or invertible if there exists a matrix B such that: worked example: • Notation. A common notation for the inverse of a matrix A is A-1. • If A is an invertible matrix, then (AT)-1 = (A-1)T • 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. 1 1 X 2 3 -1 3 = 2 + 1 3 3 -1 + 1 3 3 = 1 0 -1 2 1 3 1 3 -2+ 2 3 3 1 + 2 3 3 0 1
- 12. Determinants • Determinant is a function: – Input is nxn matrix – Output is a real or a complex number called the determinant • In MATLAB – use the command det(A)" to compute the determinant of a given square matrix A • A matrix A has an inverse matrix A-1 if and only if det(A)≠0. + + + - - -
- 13. 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
- 15. Some Application Areas • Simultaneous Equations • Simple Neural Network • GLM
- 16. System of linear equations Resolving simultaneous equations can be applied using Matrices: • Multiply a row by a non-zero constant • Interchange two rows • Add a multiple of one row to another row … Also known as Gaussian Elimination
- 17. Simplistic Neural Network O = output vector I = input vector W = weight matrix η = Learning rate d = Desired output t = time variable Given an input, provide an output… Weights learned in auto associative manner or given random values… Over time, modify weight matrix to more appropriately reflect desired behaviour
- 18. a m b3 b4 b5 b6 b7 b8 b9 + e = b + Y X × Design Matrix =
- 19. a m b3 b4 b5 b6 b7 b8 b9 + e = b + Y X × Design Matrix =
- 20. Questions?