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Linear Algebra & Matrices

Explore the basics of linear algebra and matrices, including vector operations, matrix operations, matrix inverses, and determinants. Learn about scalar, vector, and matrix definitions, as well as properties such as orthogonality and rank.

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Linear Algebra & Matrices

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  1. Linear Algebra & Matrices MfD 2004 María Asunción Fernández Seara mfseara@fil.ion.ucl.ac.uk January 21st, 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. Vector: variable described by magnitude and direction Column vector Row vector • 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 (magnitude) • Temperature = 20 °C • Density = 1 g.cm-3 • Image intensity (pixel value) = 2546 a. u.

  3. column → row row →column • Outer product = matrix Vector Operations • Transpose operator

  4. Length of a vector Right-angle triangle Pythagoras’ theorem || x || = (x12+ x22 )1/2 || x || = (x12+ x22 + x32 )1/2 Inner product of a vector with itself = (vector length)2 xTx =x12+ x22 +x32 = (|| x ||)2 x2 ||x|| x1 Vector Operations • Inner product = scalar

  5. ||x|| ||y|| b y2  q y1 x =/2 Orthogonal vectors: xTy = 0 y Vector Operations • Angle between two vectors

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

  7. Matrix Operations • 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 = BT AT Cij = inner product between ith row in A and jth column in B C = AB (m x p) = (m x n) (n x p) 2 x 33 x 22 x 2

  8. Some Definitions … • Identity Matrix • Diagonal Matrix • Symmetric Matrix I A = A I = A B = BT bij = bji

  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 (AT) -1 = (A-1)T;(A-1)T AT = (A A-1)T= I

  10. Matrix Determinant det (A) = ad - bc A (n x n) = [a ij ] 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 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. A A-1 y A z x Another Way of Looking at Matrices… • Matrix: linear transformation between two vector spaces A x = y A-1 y = x det(A) = 1 x 4 – 2 x 2 = 0 In this case, A is singular, A-1 does not exist

  13. Linearly independent Linearly dependent Other matrix definitions • Orthogonal matrix A = [q1 | q2 | … qj …| qn] qjT qq = 0 (if j  k) and qjT qj = djj AT A = D • Orthonormal matrix A = [q1 | q2 | … qj …| qn] qjT qq = 0 (if j  k) and qjT qj = 1 AT A = I A-1 = AT • Matrix rank: number of linearly independent columns or rows if rank of A (n x n) = n, then A is non-singular

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