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1 Quiz 1) Simplify ) Locate your 1 st order neighbors Name Hometown Me Name Hometown Name Hometown Name Hometown

2 Solving Linear Algebraic Equa3ons

3 Basic Concepts Here only real vector and matrices are considered. Column vector Row vector Scalar product Norm (length)

4 Basic Concepts Matrix of order Matrix A is square if m = n Matrix A consists of columns of vectors a j

5 Basic Concepts Transpose of A: (becomes matrix of order n m) Diagonal matrix:

6 Basic Concepts Identity matrix Inverse of matrix A is A -1 satisfying A is symmetric if A is orthogonal if A is square and

7 Basic Concepts Lower triangular matrix Upper triangular matrix goes analogously Exponential (A is square matrix, I is identity matrix)

8 (1) Addition Matrix Algebra (2) Subtraction For (1) and (2), A, B, and C of the same order (3) Multiplication by a number (4) Product of two matrices (in general)

9 Exponential relations Matrix Algebra

10 Trace If A is square matrix of order n n, then Trace is the sum of the diagonal elements. for order The trace of a matrix A is the sum of its eigenvalues

11 Determinants A determinant is a real number associated with every square matrix. The determinant of a square matrix A is denoted by "det A" or A. The Leibniz formula for the determinant of an n n matrix A is

12 Determinants If A is a square matrix, then the minor of the entry in the i-th row and j-th column is the determinant of the sub-matrix formed by deleting the i-th row and j-th column. The (i,j) cofactor is obtained by multiplying the minor by (-1) i+j.

13 Determinants For n = 2: For n = 3 (example, operation along the 1 st row):

14 Determinants For n = 4 (example, operation along the 2 nd column):

15 Triangular determinant Determinants

16 Determinants (7) If all elements of a row (or a column) are multiplied by a constant c, the determinant is multiplied by c. (8) Exchange of two rows (or columns) changes the sign of the determinant.

17 Determinants (9) A determinant does not change if one row (column) multiplied by a constant is added to another row (column). (10) A determinant equals zero if (i) all elements of a row (column) are zero, or (ii) two rows (columns) coincide. (11) Sub-determinant D ij of order (n-1) (n-1) is formed by deleting the i th row and the j th column of D. Cofactor

18 Matrix Inversion The inverse of a square matrix A (also called reciprocal matrix), is a matrix A 1, such that where I is the identity matrix. If matrix A has an inverse, then it is called nonsingular or, also invertible. If linearly independent. columns (rows) are

19 Matrix Inversion There are many methods to calculate A 1 : (1) Cofactor method where C is the matrix cofactors. So

20 Matrix Inversion (2) Elementary row operations from which (3) Newton s method (4) Eigendecomposition (5) Cholesky decomposition (6) etc

21 Inversion of 2 2 matrices Inversion of 3 3 matrices

22 MATLAB Implementa3ons MATLAB has built-in functions/commands for matrix operations: >> det(a) - computes matrix determinant >> inv(a) - computes matrix inverse >> trace(a) - computes matrix trace >> eig(a) - computes eigenvalues >> rank(a) - computes matrix rank >> eye(a) - generates identity matrix >> diag(a) - returns matrix diagonal elements >> sum(diag(a)) - computes matrix trace etc.

23 MATLAB Implementa3ons Creating a simple solver/function that calculates the determinant of 3 3 matrices function [d] = Det3by3(a) % Function to calculate determinant matrices size 3x3 % Operation based on the 1st row row1 = a(1,:); dd1 = a(2,2)*a(3,3) - a(2,3)*a(3,2); dd2 = a(2,1)*a(3,3) - a(2,3)*a(3,1); dd3 = a(2,1)*a(3,2) - a(2,2)*a(3,1); d = (-1)^2*row1(1,1)*dd1 + (-1)^3*row1(1,2)*dd (-1)^4*row1(1,3)*dd3;

24 MATLAB Implementa3ons To use the solver, on the Command Window: >> u=[1,1,0;4,5,9;0,2,8] u = >> Det3by3(u) ans -10

25 Systems of Linear Equa3ons The general form of systems of linear equations (3.1) where a s are constant coefficients, b s are constants, m is the number of equations, and n is the number of variables or unknowns. The system (3.1) is homogeneous if all otherwise inhomogeneous.

26 Systems of Linear Equa3ons Arranging in matrix and vector notation or (3.2) The matrix A is called the coefficient matrix and has size (m rows and n columns).

27 Systems of Linear Equa3ons Solutions of general linear systems with m equations in n unknowns may be classified into exactly three possibilities: (1) No solution (2) Infinitely many solutions (3) A unique solution

28 Solving Equa3ons with Geometry If n = 2 (plane) or n = 3 (space), the system of equations (3.1) has a geometric interpretation that can give us valuable intuition about possible solutions. Plane Geometry - represented by line equations (3.3) Solving the system above is the geometrical equivalent of finding all possible (x,y)-intersections of the lines.

29 Solving Equa3ons with Geometry Parallel lines, no solution Identical lines, infinitely many solutions Non-parallel distinct lines, one solution at the unique intersection

30 Solving Equa3ons with Geometry Space Geometry - A plane in xyz-space is given by plane equation The vector is normal to the plane. (3.4) Solving the system above is the geometrical equivalent of finding all possible (x,y,z)-intersections of the planes.

31 Solving Equa3ons with Geometry A B C A=B Triple-decker. Planes A, B, and C are parallel. No intersection point. Double-decker. Planes A and B are equal and parallel to plane C. No intersection. C C A B Book shelf. Planes A and B are distinct and parallel. No intersection point.

32

33 Solving Equa3ons with Matrix Op. A system of m linear equations and n unknowns can be written in matrix notation as: or solving it: (3.5) (3.6)

34 Solving Equa3ons with Matrix Op. If m=n in Eq. (3.1) and if det A 0, the unique solution can be expressed explicitly by either: where det A is the determinant of matrix A, det A j is the determinant that arises when the j-th column of det A is replaced by the column elements b 1,, b n (Cramer s rule).

35 MATLAB Implementa3ons Solve the following system of equations using MATLAB: >> A = [0.3,0.52,1;0.5,1,1.9;0.1,0.3,0.5]; >> b = [-0.01, 0.67, -0.44] ; >> x = A\b x =

36 Solving of Equa3ons by Gaussian Elimina3on If m n in Eq. (3.1), Gaussian elimination can be employed to solve the system of equations (3.1). By elementary row operations (i.e., (i) exchange of equations, (ii) multiplication of an equation by a constant, (iii) adding one equation to another), the system (3.1) can be transformed into echelon form. From the echelon form, the unknowns n are determined by backward substitution. Of these there are 2 groups: 1) Basic variables: corresponding to pivots 2) Free variables: the others

37 Solving of Equa3ons by Example: Gaussian Elimina3on x,y,z are basic variables, u is free variable.

38 Eigenvalues The number λ is called an eigenvalue of a square matrix A and g 0 is a corresponding eigenvector if (3.7) Characteristic equation is the equation which is solved to find a matrix s eigenvalues. The characteristic equation in variable λ is defined by

39 Eigenvalues (product of all eigenvalues) (sum of eigenvalues) (Real square) matrix A is symmetric if If matrix A is symmetric: (1) all eigenvalues are real (2) eigenvectors corresponding to different eigenvalues are orthogonal (3)

40 Eigenvalues The characteristic equation of a 2x2 matrix: or can be written in the particular form: Similarly, the characteristic equation of a 3x3 matrix:

41 Eigenvalues or, can be written as well as where tr(a) is the matrix trace of A and det(a) is its determinant.

42 Spectral Theorem Assume that: (1) Matrix A is symmetric (2) are its eigenvalues Then there exist corresponding eigenvectors g 1,, g n such that: (1) (pairwise orthogonal and normed) Set, then (2) P is an orthogonal matrix (3) (4) (5)

43 Eigenvalues & Eigenvectors Example: Find eigenvalues and eigenvectors of matrix Solution The characteristic equation of matrix A is with roots

44 Eigenvalues & Eigenvectors Using Eq. (3.3), which states and we get If we choose the general solutions

45 Eigenvalues & Eigenvectors then the unknown vector is given by Since eigenvectors are normalized, then From

46 Eigenvalues & Eigenvectors we get If we choose the general solutions then the second unknown vector is given by

47 Eigenvalues & Eigenvectors From Choosing the general solutions the third eigenvector is given by

48 MATLAB Implementa3ons Matlab commands for matrix operations: >> A=[8,1,6;3,5,7;4,9,2] >> B = A >> trace(a) >> rank(a) >> det(a) >> inv(a)

49 MATLAB Implementa3ons Creating a diagonal matrix from a vector: >> A=[1,2,4,5,7,9]; >> B=diag(A) B =

50 MATLAB Implementa3ons Generating a vector: >> A=zeros(1,2) A = 0 0 Transpose of a vector: >> B=A B = 0 0 Creating a 2 3 matrix: >> D=[1,2,4;5,7,9] D = Multiplication of matrices/vectors: >> C=A.*B; Suppress output

51 MATLAB Implementa3ons Generating a 5 5 matrix whose elements are all zeros: Generating a 5 3 matrix whose elements are all ones >> A=zeros(5) A = >> B=ones(5,3) B =

52 MATLAB Implementa3ons Concatenating matrices A and B horizontally : >> C=[A,B] C = (Same length of rows) Concatenating matrices vertically: >> C=[A;B] (Same length of columns)

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