Download presentation

Presentation is loading. Please wait.

Published byPreston Horn Modified over 4 years ago

1
Linear Algebra & Matrices MfD 2004 María Asunción Fernández Seara mfseara@fil.ion.ucl.ac.uk January 21 st, 2004 “The beginnings of matrices and determinants go back to the second century BC although traces can be seen back to the fourth century BC”

2
Scalar: variable described by a single number (magnitude) –Temperature = 20 °C –Density = 1 g.cm -3 –Image intensity (pixel value) = 2546 a. u. Scalars, Vectors and Matrices Column vector Row vector 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

3
Vector Operations Transpose operator column → rowrow → column Outer product = matrix

4
Vector Operations Inner product = scalar || x || = (x 1 2 + x 2 2 ) 1/2 || x || = (x 1 2 + x 2 2 + x 3 2 ) 1/2 Inner product of a vector with itself = (vector length) 2 x T x =x 1 2 + x 2 2 +x 3 2 = (|| x ||) 2 x1x1 x2x2 ||x|| Right-angle triangle Pythagoras’ theorem Length of a vector

5
Vector Operations Angle between two vectors Orthogonal vectors: x T y = 0 x y = /2 ||x|| ||y|| y2y2 y1y1

6
Addition (matrix of same size) –Commutative: A+B=B+A –Associative: (A+B)+C=A+(B+C) Matrix Operations

7
Multiplication (number of columns in first matrix = number of rows in second) –Associative: (A B) C = A (B C) –Distributive: A (B+C) = A B + A C –Not commutative: AB BA!!! –(A B) T = B T A T Matrix Operations 2 x 3 3 x 2 2 x 2 C = AB (m x p)= (m x n) (n x p) C ij = inner product between i th row in A and j th column in B

8
Some Definitions … Identity Matrix Diagonal Matrix Symmetric Matrix I A = A I = A B = B T b ij = b ji

9
Matrix Inverse A -1 A = A -1 A = I Properties A -1 only exists if A is square (n x n) If A -1 exists then A is non-singular (invertible) (A B) -1 = B -1 A -1 ; B -1 A -1 A B = B -1 B = I (A T ) -1 = (A -1 ) T ; (A -1 ) T A T = (A A -1 ) T = I

10
Matrix Determinant det (A) = ad - bc Properties Determinants are defined only for square matrices If det(A) = 0, A is singular, A -1 does not exist If det(A) 0, A is non-singular, A -1 exists A (n x n) = [a ij ] http://mathworld.wolfram.com/Determinant.html

11
Matrix Inverse - Calculations A general matrix can be inverted using methods such as the Gauss-Jordan elimination, Gauss elimination or LU decomposition

12
Another Way of Looking at Matrices… Matrix: linear transformation between two vector spaces A x = y A -1 y = x x y A A -1 z A det(A) = 1 x 4 – 2 x 2 = 0 In this case, A is singular, A -1 does not exist

13
Other matrix definitions Linearly independentLinearly dependent Orthonormal matrix A = [q 1 | q 2 | … q j …| q n ] q j T q q = 0 (if j k) and q j T q j = 1 A T A = I A -1 = A T Matrix rank: number of linearly independent columns or rows if rank of A (n x n) = n, then A is non-singular Orthogonal matrix A = [q 1 | q 2 | … q j …| q n ] q j T q q = 0 (if j k) and q j T q j = d jj A T A = D

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google