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