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1 Knowledge Discovery and Data Mining 1 (VO) ( ) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
2 Big picture: KDDM Probability Theory Linear Algebra Hardware & Programming Model Information Theory Statistical Inference Mathematical Tools Infrastructure Knowledge Discovery Process Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
3 Outline 1 Matrices and Vectors 2 Matrix Multiplication 3 Operations and Properties 4 Eigenvalues and Eigenvectors Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
4 Matrices and Vectors Matrix Matrix Matrix is a rectangular array of numbers, i.e. A R m n a 11 a a 1n a 21 a a 2n A = a m1 a m2... a mn Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
5 Matrices and Vectors Matrix: Examples A = A R 2 3 B R 4 2 ( ) m and n are matrix dimensions B = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
6 Matrices and Vectors Multiplication with a scalar Scalar multiplication Given a matrix A R m n and a scalar α R the scalar multiplication of A with α is the matrix B = αa R m n, where B ij = αa ij αa 11 αa αa 1n αa 21 αa αa 2n αa = αa m1 αa m2... αa mn Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
7 Matrices and Vectors Multiplication with a scalar: example 2 ( ) = ( ) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
8 Matrices and Vectors Vector Vector Vector is a matrix consisting of one column, i.e. x R n. A row vector is a matrix consisting of one row. x 1 x 2 x =. x n Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
9 Matrices and Vectors Vector: Example x = x R 4 n is the vector dimension Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
10 Matrix Multiplication Matrix Multiplication Matrix Multiplication The product of two matrices A R m n and B R n p is the matrix C = AB R m p, where n C ij = A ik B kj (1) k=1 For the product to exist: the number of columns in A must equal the number of rows in B Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
11 Matrix Multiplication Matrix Multiplication: Example ( ) 4 5 ( ) a11 a = a a 22 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
12 Matrix Multiplication Matrix Multiplication: Example ( ) 4 5 ( ) a11 a = a a 22 a 11 = 1 4+ Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
13 Matrix Multiplication Matrix Multiplication: Example ( ) 4 5 ( ) a11 a = a a 22 a 11 = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
14 Matrix Multiplication Matrix Multiplication: Example ( ) 4 5 ( ) a12 = a a 22 a 11 = = 30 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
15 Matrix Multiplication Matrix Multiplication: Example ( ) = ( ) a 21 a 22 a 12 = = 39 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
16 Matrix Multiplication Matrix Multiplication: Example ( ) = ( ) a 22 a 21 = = 34 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
17 Matrix Multiplication Matrix Multiplication: Example ( ) = ( 30 ) a 22 = = 29 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
18 Matrix Multiplication Vector-Vector products Inner product Given two vectors x, y R n the quantity x T y, sometimes called the inner product or dot product, is a real number given by x T y = ( y 1 ) x 1 x 2... x n y 2 n = x i y i.y i=1 n The inner products are just special cases of matrix multiplication Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
19 Matrix Multiplication Inner product: Example ( ) = x T y = = 29 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
20 Matrix Multiplication Vector-Vector products Outer product Given two vectors x R m and y R n the matrix xy T is called the outer product of the vectors. The elements of the matrix are given by (xy T ) ij = x i y j, i.e. x 1 x 1 y 1 x 1 y 2... x 1 y n xy T x 2 ( ) x 2 y 1 x 2 y 2... x 2 y n = y1 y 2... y n = x m x m y 1 x m y 2... x m y n Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
21 Matrix Multiplication Matrix-Vector products Matrix-vector product Given a matrix A R m n and a vector x R n, their product is a vector y = Ax R m, i.e. a 11 a a 1n x n 1 i=1 a 21 a a 2n x 2 a n 1ix i a T Ax = = i=1 a 1 2ix i x. = a T 2 x. a m1 a m2... a mn x n n i=1 a mix i a T mx Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
22 Matrix Multiplication Matrix-Vector products: Example ( ) = ( ) 30 y 2 y 1 = = 30 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
23 Matrix Multiplication Matrix-Vector products: Example ( ) = ( ) y 2 = = 34 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
24 Matrix Multiplication Matrix-Vector products Matrix-vector product Given a matrix A R m n and a vector x R m, their (left row) product is a vector y t = x T A R n, i.e. x T A = = a 11 a a 1n a ( ) 21 a a 2n x1 x 2... x m a m1 a m2... a mn ( mi=1 a i1 x mi=1 i a i2 x i... mi=1 ) ) a in x i = (x T a 1 x T a 2... x T a m Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
25 Matrix Multiplication Matrix-Vector products: Example ( ) ( ) = ( ) 34 y y 3 y 1 = = 34 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
26 Matrix Multiplication Matrix-Vector products: Example ( ) ( ) = ( ) y y 2 = = 28 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
27 Matrix Multiplication Matrix-Vector products: Example ( ) ( ) = ( ) y 3 = = 22 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
28 Matrix Multiplication Properties of matrix multiplication Matrix multiplication is associative: (AB)C = A(BC) Matrix multiplication is distributive: A(B + C) = AB + AC Matrix multiplication is not commutative: AB BA Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
29 Matrix Multiplication Matrix Sum Matrix Sum The sum of two matrices A R m n and B R m n is the matrix C = A + B R m n, where C ij = A ij + B ij (2) For the sum to exist: the dimension of A and B must be equal Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
30 Matrix Multiplication Matrix Sum: Example ( ) ( ) = ( 5 7 ) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
31 Matrix Multiplication Linear combinations of vectors Linear combination Given are k vectors x i R n and k scalars α i R. Then the linear combination of those vectors with those scalars is defined as: α 1 x 1 + α 2 x α k x k = k α i x i (3) i=1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
32 Matrix Multiplication Linear combinations of vectors: example = (4) In fact, any vector in R 3 is a linear combination of those three vectors Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
33 Operations and Properties Identity matrix Identity matrix The identity matrix, denoted I R n n is a square matrix with ones on the diagonal and zeros everywhere else, that is { 1, i = j I ij = 0, i j The identity matrix has the property that for all A R m n AI = A = IA (5) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
34 Operations and Properties Identity matrix I = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
35 Operations and Properties Diagonal matrix Diagonal matrix A diagonal matrix, denoted D R n n is a square matrix with all non-diagonal elements zero. This is typically denoted D = diag(d 1, d 2,..., d n ), with { d i, i = j D ij = 0, i j The identity matrix is then I = diag(1, 1,..., 1) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
36 Operations and Properties Diagonal matrix D = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
37 Operations and Properties The Transpose Transpose The transpose of a matrix results from flipping the rows and columns. Given a matrix A R m n, its transpose A T R n m, is the n m matrix whose entries are given = A ji A T ij We have been using the transpose for describing row vectors Transpose of a column vector is a row vector Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
38 Operations and Properties Properties of transpose (A T ) T = A (AB) T = B T A T (A + B) T = A T + B T Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
39 Operations and Properties Transpose: Example ( ) T = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
40 Operations and Properties Symmetric Matrices Symmetric matrices A square matrix A R n n is symmetric if A = A T. For any matrix A R n n, the matrix AA T is symmetric Proof left for exercise ;) Symmetric matrices occur very often in practice and they have many nice properties Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
41 Operations and Properties Symmetric Matrices: Example A = (6) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
42 Operations and Properties Trace Trace The trace of a square matrix A R n n, denoted by tr(a) is the sum of diagonal elements in the matrix: tr(a) = n A ii (7) i=1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
43 Operations and Properties Trace: example tr(a) = tr( ) = 2 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
44 Operations and Properties Norms Norm A norm of a vector x is informally a measure of the length of the vector. For example, we have the commonly used Euclidean or l 2 norm, Note that: x 2 2 = xt x x 2 = n xi 2 (8) i=1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
45 Operations and Properties Norms Norm A norm is any function f : R n R that satisfies 4 properties: 1 For all x R n, f (x) 0 (non-negativity) 2 f (x) = 0, if and only if x = 0 (definiteness) 3 For all x R n, t R, f (tx) = t f (x) (homogeneity) 4 For all x, y R n, f (x + y) f (x) + f (y) (triangle inequality) Norms are used to normalize vectors and obtain unit vectors (vectors of length 1) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
46 Operations and Properties Norms Other examples of norms include l 1, l, or more generally l p, with p 1 x 1 = n x i i=1 x = max i x i ( n ) 1/p x p = x i p i=1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
47 Operations and Properties Norms: example 4 x = 10 2 x 1 = 16 x 2 = x = 10 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
48 Operations and Properties Linear independence Linear independence A set of vectors {x 1, x 2,..., x k } R n is said to be linearly independent if no vector can be represented as a linear combination of the remaining vectors. Linear independence A set of vectors {x 1, x 2,..., x k } R n is said to be linearly independent if no scalars {α 1, α 2,..., α k } exist such that k α i x i = 0 i=1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
49 Operations and Properties Linear independence: example x 1 = 0 x 2 = 1 x 3 = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
50 Operations and Properties Linear dependence Conversly, if one vector belonging to the set can be represented as a linear combination of the remaining vectors the vectors are linearly dependent x 1 = 2 x 2 = 1 x 3 = Dependent because: x 3 = 2x 1 + x 2 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
51 Operations and Properties Vector bases Span A set of n vectors {x 1, x 2,..., x n } R n is said to span R n, i.e. they form a basis for the space. Any vector v R n can be written as a linear combination of x 1 through x n. Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
52 Operations and Properties Vector bases Orthogonal basis An orthogonal basis {x 1, x 2,..., x n } R n satisfies x T i x j = 0, if i j Orthonormal basis An orthonormal basis {x 1, x 2,..., x n } R n satisfies x T i x j = 0, if i j x T i x j = 1, if i = j Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
53 Operations and Properties Orthonormal basis: example x 1 = 0 x 2 = 1 x 3 = Orthonormal basis for R 3 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
54 Operations and Properties Rank Rank The column rank of a matrix A R m n is the size of the largest subset of columns that constitute a linearly independent set. The row rank is the largest subset of rows that constitute a linearly independent set. For any matrix A R m n the columns rank is equal to row rank and both quantities are referred to as the rank of A, denoted as rank(a) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
55 Operations and Properties Rank: example rank( ) = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
56 Operations and Properties Properties of the matrix rank Some basic properties of the rank: 1 For A R m n, rank(a) min(m, n). If rank(a) = min(m, n) then A is said to be full rank. 2 For A R m n, rank(a) = rank(a T ) 3 For A R m n, B R n p, rank(ab) = min(rank(a), rank(b)) 4 For A, B R m n, rank(a + B) rank(a) + rank(b) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
57 Operations and Properties The Inverse Inverse The inverse of a square matrix A R n n is denoted A 1 and is a unique matrix such that A 1 A = I = AA 1 Not all matrices have inverses If A 1 exists we say that A is invertible or non-singular If A 1 does not exist we say that A is non-invertible or singular Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
58 Operations and Properties The Inverse For A to be invertible it must be full rank There are many alternative conditions for invertability The inverse is used in e.g. solving the system of linear equations: Ax = b With A R n n and x, b R n Then x = A 1 b Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
59 Operations and Properties Properties of inverse (A 1 ) 1 = A (AB) 1 = B 1 A 1 (A 1 ) T = (A T ) 1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
60 Operations and Properties Inverse: Example ( ) 1 a b = c d 1 ad bc ( ) = ( d ) b c a ( 5 ) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
61 Operations and Properties Orthogonal matrices Inverse A square matrix U R n n is orthogonal if all its columns are orthonormal (orthogonal to each other and normalized). It follows from the definition: U T U = I = UU T In other words, the inverse of an orthogonal matrix is its transpose Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
62 Operations and Properties Orthogonal matrices: Example ( ) T ( ) = ( ) 1 0 = 0 1 ( ) ( ) T Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
63 Operations and Properties The determinant Determinant The determinant is a real number associated with a square matrix A R n n, denoted det(a). Depending on n it can be calculated by different arithmetic expressions. ( ) a b det( ) = ad bc c d ( ) 6 2 det( ) = Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
64 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors Eigenvalues and eigenvectors Given a square matrix A R n n, we say that λ C is an eigenvalue of A and x C n is the corresponding eigenvector if Ax = λx, x 0 Intuitively, multiplying A by the vector x does not change the direction of x The new vector points in the same direction but is scaled by factor λ Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
65 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors Also, for any eigenvector x C n, and scalar c C: A(cx) = cax = cλx = λ(cx) Thus, cx is also an eigenvector with eigenvalue λ For this reason, we usually normalize the eigenvector x to have length 1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
66 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors We can rewrite the equation from above (it holds: x 0): Ax = λx Ax = λix λix Ax = 0 (λi A)x = 0 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
67 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors (λi A)x = 0, x 0 has a non-zero solution only if the matrix (λi A) is singular, i.e.: det(λi A) = 0 This can be expanded in a very large polynomial in λ, with maximal degree n We then find the roots of that polynomial and obtain the eigenvalues λ 1,..., λ n Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
68 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors To find the eigenvector corresponding to the eigenvalue λ i we simply solve the linear equation: (λ i I A)x = 0 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
69 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors: example ( ) 2 4 A = 1 1 ( ) λ 2 4 det(λi A) = det( ) = (λ 2)(λ + 1) 4 1 λ + 1 = λ 2 2λ + λ 2 4 = λ 2 λ 6 = (λ 3)(λ + 2) Thus, λ 1 = 3, and λ 2 = 2 are eigenvalues of A We now solve (λ i I A)x = 0 for each eigenvalue to find the corresponding eigenvectors Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
70 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors: example For λ 1 = 3 ( ) ( ) 1 4 x1 = 1 4 x 2 ( ) 0 0 x 1 + 4x 2 = 0 x 1 + 4x 2 = 0 Thus, x 1 = 4x 2, and we might pick x = And we normalize to: ( ) 4/ 17 1/ 17 ( ) 4 1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
71 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors: example For λ 1 = 2 ( ) ( ) 4 4 x1 = 1 1 x 2 ( ) 0 0 4x 1 + 4x 2 = 0 x 1 x 2 = 0 Thus, x 1 = x 2, and we might pick x = And we normalize to: ( ) 1/ 2 1/ 2 ( ) 1 1 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
72 Eigenvalues and Eigenvectors Properties of eigenvalues and eigenvectors For A R n n, eigenvalues λ 1,..., λ n, and associated eigenvectors x 1,..., x n the following are properties: 1 The trace of A is equal to the sum of its eigenvalues: tr(a) = 2 The determinant of A is equal to the product of its eigenvalues: det(a) = n i=1 n i=1 λ i λ i Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
73 Eigenvalues and Eigenvectors Properties of eigenvalues and eigenvectors For A R n n, eigenvalues λ 1,..., λ n, and associated eigenvectors x 1,..., x n the following are properties: 3 The rank of A is equal to the number of non-zero eigenvalues of A 4 If A is non-singular then 1/λ i is an eigenvalue of A 1 with associated vector x i, i.e. A 1 x i = (1/λ i )x i 5 The eigenvalues of a diagonal matrix D = diag(d 1,..., d n ) are just the diagonal entries d 1,..., d n Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
74 Eigenvalues and Eigenvectors Eigenvalues and eigenvectors of symmetric matrices There are two important properties of eigenvalues and eigenvectors of symmetric matrices All eigenvalues are real The eigenvectors are orthonormal If the eigenvectors are linearly independent then we can decompose A as UΛA T U is an orthogonal matrix where colums are the eigenvectors Λ = diag(λ 1,..., λ n ) Denis Helic (KTI, TU Graz) KDDM1 Oct 9, / 74
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