Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014

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1 Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014

2 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of linear equation, eigenvectors.

3 Course overview 11 lectures of 3 4 hours. Examination consists of 2 exercise assignments one project with presentation.

4 Changes from previous years Greater focus on multivariate analysis techniques. Mainly in order to accomodate for the changes the course compendium will be updated during the course.

5 Repetition matrices A matrix is a rectangular array of numbers, symbols or expressions. Some example of matrices can be seen below: A = [ ] B = 1 2 C = 2 3 D = 3 1 [ ] 1 + i i [ 1 0 ] Note that matrices need not be square as in C D.

6 Repetition matrices: notation 1 2 We take a look at matrix C = We denote every number or rather element of A as a i,j, where i is it s row number j is it s column number. What is c 3,1 of C above? The size of a matrix is the number of rows columns (or the index of the bottom right element): We say that C above is a 3 2 matrix. A matrix with only one row or column is called a row vector or column vector respectively. The diagonal of a matrix is the elements diagonally from the top left corner towards the bottom left (a 1,1, a 2,2,... a n,n ). A matrix where all elements not on the diagonal is zero is called a diagonal matrix.

7 Repetition matrices: addition subtraction Add every element in the first matrix with correspondign element in the second matrix. We dem that both matrices have the same size. [ ] [ ] 1 2 = 2 1 [ ] = [ ]

8 Repetition matrices: multiplication Given the two matrices A B we get the product AB as the matrix whose elements e i,j are found by multiplying the elements in row i of A with the elements of column j of B adding the results = For the colored element we have: = 5

9 Repetition matrices: multiplication We note that we need the number of columns in A to be the same as the number of rows in B but A can have any number of rows B can have any number of columns. Also we have: Generally AB BA, why? The size of AB is the number of rows of A times the number of columns of B. The Identity matrix I is the matrix with ones on the diagonal zero elsewere. For the Identity matrix we have: AI = IA = A.

10 Repetition matrices: multiplication Multiplying a scalar with a matrix is done by multiplying the scalar with every element of the matrix. 3 [ ] 1 0 = 0 2 [ ] = [ ]

11 equation matrices equation system a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.. x 1. x a m1 x 1 + a m2 x a mn x n = b m Represented with a matrix column vectors a 11 a a 1n x 1 b 1 a 21 a a 2n x = b 2. a m1 a m2... a mn x m b m

12 Solving a linear equation system Gaussian elimination. Elementary row operations (adding multiplying rows). Find pivot elements, change matrix to diagonal/echelon row form

13 When can a linear system be solved? From previous courses: Theorem The following statements are equivalent for A M n n (K): 1. AX = B has a unique solution for all B M n m where X M n m. 2. AX = 0 only has the trivial solution (X = 0). 3. A has linearly independent row/column vectors. 4. A is invertible/non-degenerate/non-singular. 5. det(a) A has maximimum rank (rank(a) = n). 7. A s row/column space has dimension n. 8. A s null space has dimension A s row/column vectors span K

14 Definition (1) For A M n n (K) the rank(a) is the number of linearly independent rows or columns. To see that the number or linearly independent rows is the same as the number of linearly independent columns, draw a matrix in echelon form. Definition (2) For A M n n (K) the rank(a) is dimension of the column space of A.

15 What do we already know about determinants? Let A, B M n n ((K)) Know how to calculate it for 1 1, matrices. Know how to calculate it for triangular matrices matrices with linearly dependent rows. det A = det A det AB = det A det B but det(a + B) det(a) + det(b). det ca = c n det A... plus a few more things.

16 Calculating determinants using cofactors Theorem det(a) = where A ki is the ki-cofactor of A. n a ki A ki k=1 The ki-cofactor is ( 1) i+k times the determinant of A after we have removed row k column i.

17 Definition The inverse of a matrix A M n n is a matrix X M n n such that AX = XA = I The inverse is denoted X = A 1. If AX = B then A 1 AX = A 1 B IX = X = A 1 B thus we can solve linear equation if we know the inverse.

18 What do we know about inverses? (A 1 ) 1 = A (A + B) 1 A 1 + B 1 (AB) 1 = B 1 A 1 (ab) 1 = 1 a B 1 (A 1 ) = (A ) 1

19 Calculate the inverse Use gaussian elimination or the adjugate formula A 1 = 1 det(a) adj(a) where adj(a) is the adjugate matrix. Definition The adjugate matrix adj(a) of a matrix is the transpose of the cofactor matrix where we have (adj(a)) ik = A ki.

20 Other kinds of inverses Left right inverses BA = I, AB = I Pseudoinverses ABA = A BAB = B These kinds inverses can exist for non-square matrices, A M m n, B M n m

21 Eigenvalues eigenvectors Definition If Av = λv where A M n n (K), λ K, v M n 1 (K) then λ is an eigenvalue of A v is an eigenvector if v 0.

22 What do we know about eigenvectors? Can have several different eigenvectors. Different eigenvectors can have the same eigenvalue. Can find the by solving det(a λi) = 0 the eigenvectors by solving (A λi)v = 0.

23 Spectrum of a matrix Set of for a matrix A is called the spectrum of A is denoted by Sp(A). The absolute value of the largest eigenvalue is called the spectral radius ρ(a) = max λ λ Sp(A)

24 Finding the Av = λv (A λi)v = 0 if v 0 then det(a λi) = 0. Many different methods, for example Gaussian elimination. Can find the by solving det(a λi) = 0 which is an nth degree polynomial in λ. p A (λ) = det(a λi) is called the characteristic polynomial p A (λ) = 0 is called the characteristic equation.

25 Important results with tr(a) = det(a) = n a i,i = i=1 n i=1 λ i n i=1 λ i If A M n n (K) have n different then A is diagonalizable.

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