University of Nottingham

1. School of Economics University of Nottingham Pre-sessional Mathematics Masters Dr Maria Montero Dr Alex Possajennikov Topic 2 Linear Algebra Pre-sessional Mathematics Topic 2

2. An m  nmatrix is an rectangular array of numbers with m rows and n columns: Linear Algebra Matrices denotes the entry in the ith row and jth column Pre-sessional Mathematics Topic 2

3. Linear Algebra Vectors A column vector is a matrix with n rows and 1 column A row vector is a matrix with 1 row and n columns Pre-sessional Mathematics Topic 2

4. Linear Algebra Some properties of matrices Symmetric: Square: aji = aij m= n Pre-sessional Mathematics Topic 2

5. Linear Algebra Triangular: aij = 0when j >i aij = 0when j < i Identity: Diagonal: aii = 1 aij = 0when j  i aij = 0when j  i Pre-sessional Mathematics Topic 2

6. Linear Algebra Sum of matrices of the same dimension: Multiplication by scalar: Transpose of a m  n matrix is a n  m matrix Transpose: Pre-sessional Mathematics Topic 2

7. Scalar product (of vectors) The product of a row vector a and a column vector b is a scalar a  b = a1b1 + ... + anbn Pre-sessional Mathematics Topic 2

8. Linear Algebra The product of two matrices A and B is another matrix C cijis the scalar product of the i-th row of A and the j-th column of B Example: The product of an m  n matrix A and n  p matrix B is an m  p matrix Identity matrix The product of two matrices is not (generally) commutative Pre-sessional Mathematics Topic 2

9. Linear Algebra Properties of matrix product AB  BA (AB)C = A(BC) A(B + C) = AB + AC (AB)’ = B’A’ Length of a vector Orthogonal vectors a’b = 0 Pre-sessional Mathematics Topic 2

10. Squaren  n matrices The inverse of n  nmatrix A is n  n matrix A-1 such that AA-1 = A-1A = In If inverse A-1 exists then Ais non-singular If inverse A-1 does not exist then Ais singular Example: If a11a22 - a12a21 = 0, then A-1 does not exist (A-1)-1 = A (A-1)’ = (A’)-1 If A is symmetric, then A-1 is symmetric (AB)-1 = B-1A-1 Pre-sessional Mathematics Topic 2

11. Linear Algebra Systems of linear equations n equations m variables Ax = b Suppose n = mand A-1 exists A-1(Ax) = A-1b x = A-1b Pre-sessional Mathematics Topic 2

12. Linear Algebra Determinant Squaren  n matrices Let Aij be the (n - 1)  (n - 1)matrix obtained by deleting row i and column j For 1  1 matrix A = |a11|, det A = a11 For n  n matrix A, Example Pre-sessional Mathematics Topic 2

13. Linear Algebra Properties of the determinant det(AB) = det A ·det Bdet A’ = det Adet A-1 = 1/det(A) If A is triangular, det A = a11·a22 ·... ·ann Theorem If det A  0, then A is non-singular Cij = (-1)i+jAijis called cofactor of aij Transpose of matrix of cofactors is called adjointadj(A)of A Example Pre-sessional Mathematics Topic 2

14. Linear Algebra Determinant and systems of linear equations x = A-1b Example Trace of a matrix is the sum of the diagonal elements: tr A = a11 + … + ann Pre-sessional Mathematics Topic 2

15. Linear Algebra Eigenvalues (characteristic roots) Squaren  n matrices Ax = x Ax = Ix Ax - Ix = 0 (A - I)x = 0 If (A - I)-1exists, then x = (A - I)-10 = 0 (A - I)-1does not exist det (A - I) = 0 Characteristic polynomial of A If det (A - I) = 0, then  is an eigenvalue of A If Ax = x when  is an eigenvalue of A, then x is an eigenvector of A Since characterisic polynomial has ordern, there are usually neigenvalues and neigenvectors Pre-sessional Mathematics Topic 2

16. Linear Algebra Example If x is an eigenvector, then x is also an eigenvector Normalised eigenvector: ||x|| = 1 Pre-sessional Mathematics Topic 2

17. P = (x1 … xn) Linear Algebra Diagonalisation (spectral decomposition) of a matrix 1,…,n are eigenvalues of A, x1,…,xn are eigenvectors of A Since Ax = x, AP = PD P-1AP = D and A = PDP-1 Suppose P-1 exists Not every matrix can be diagonalised If A is symmetric, then A has nreal eigenvalues and the eigenvectors are orthogonal P-1 = P’ when eigenvectors are normalised A = PDP’ det A = det D = 1·…·ntr A = trD = 1 + … + n Pre-sessional Mathematics Topic 2

18. Linear Algebra Example Pre-sessional Mathematics Topic 2

19. symmetric Linear Algebra Quadratic forms and their definiteness q = a11x1x1 + a12x1x2 + … + annxnxn q = x’Ax Matrix A is positive definite if x’Ax > 0 for all x  0 Matrix A is negative definite if x’Ax < 0 for all x  0 Matrix A is positive semidefinite if x’Ax  0 for all x  0 Matrix A is negative semidefinite if x’Ax  0 for all x  0 If none of the above - indefinite Pre-sessional Mathematics Topic 2

20. Linear Algebra If all eigenvalues of A are positive, then A is positive definite If all eigenvalues of A are negative, then A is negative definite Leading principal minors A1 = a11 A3 = A If det Ai > 0 for all i, then A is positive definite If (-1)idet Ai > 0 for all i, then A is negative definite Example Pre-sessional Mathematics Topic 2