Linear Algebra Massoud Malek


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1 CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product is a generalization of the dot product In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar More precisely, for a real vector space, an inner product, satisfies the following four properties Let u, v, and w be vectors and α be a scalar, then: Linearity: u + v, w = u, w + v, w α v, w = α v, w Symmetry: v, w = w, v Positivedefiniteness: v, v 0 and equal if and only if v = 0 A vector space together with an inner product on it is called an inner product space When given a complex vector space, the symmetry property above is usually replaced by Conjugate Symmetry: v, w = w, v, where w, v refers to complex conjugation With this property, the inner product is called a Hermitian inner product and a complex vector space with a Hermitian inner product is called a Hermitian inner product space An inner product space is a vector space with the additional inner product structure An inner product naturally induces an associated norm, u = u, u Thus an inner product space is also a normed vector space Examples of Inner Product Space (a) The Euclidean inner product on IF n, where IF is the set of real or complex number is defined by (u, u 2,, u n, v, v 2,, v n ) = u v + u 2 v u n v n (b) An inner product can be defined on the vector space of smooth realvalued functions on the interval [, ] by f)x), g(x) = f(x) g(x) d x
2 Linear Algebra Inner Product and Normed Space Page 2 (c) An inner product can be defined on the set of real polynomials by Examples of Vector Norms p(x), q(x) = Here are three mostly used vector norms: (a) norm: v = v i i (b) 2norm or Euclidean norm: v 2 = v i 2 (c) norm: v = max x i 0 p(x) q(x) e x d x i Also, note that if is a norm and M is any nonsingular square matrix, then v M v is also a norm The case where M is diagonal is particularly common in practice Note Any Hermitian positive definite matrix H induces an inner product u, v H = u H v; and thus a norm u H = u, u H In MatLab, the norm, 2norm, and norm are invoked by the statements norm (u, ), norm (u, 2), and norm (u, inf), respectively The 2norm is the default in MatLab The statement norm (u) is interpreted as the 2norm of u in MatLab Theorem If is a norm induce by an inner product, then u = 0, if and only if u = θ, otherwise u > 0 ; 2 λ u = λ u ; 3 (CauchySchwarz inequality) u, v u v ; 4 (Triangle inequality) u + v u + v, Orthogonality In an inner product vector space V, vectors u and v are orthogonal, if u, v = 0 A subset S of V is orthogonal, if any two distinct vectors of S are orthogonal A vector u is a unit vector, if u = Finally a subset S of V is orthonormal, if S is orthogonal and consists entirely of unit vectors If v θ, then the vector u = v is a unit vector; we say that v was normalized v Note The zero vector θ is orthogonal to every vector in V and θ is the only vector in V that is orthogonal to itself Pythagorean Theorem Suppose u and v are orthogonal vectors in V Then Proof we have u + v 2 = u 2 + v 2 u + v 2 = u + v, u + v = u, u + u, v + v, u + v, v = u, u 2 + v, v 2 as desired
3 Linear Algebra GramSchmidt Page 3 The CauchySchwarz inequality states that for all vectors u and v of an inner product space: u, v 2 u, u v, v, where, is the inner product Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as u, v u v Moreover, the two sides are equal if and only if u and v are linearly dependent Proof Let u, v be arbitrary vectors in a vector space V over IF with an inner product, where IF is the field of real or complex numbers If u, v = 0, the theorem holds trivially If not, then u 0, v 0 For any λ IF, we have 0 u λ v 2 = u λ v, u λ v = u, u λ v λ v, u λ v u, u λ u, v λ v, u + λ λ v, v Choose λ = u, v v, v, then 0 u, u λ u, v λ v, u + λ λ v, v = u, u u, v 2 v, v = u 2 u, v 2 v 2 It follows that u, v 2 u, u v, v or u, v u v Examples of the CauchySchwarz Inequality (a) If u, u 2,, u n, v, v 2,, v n are real numbers, then u v + u 2 v u n v n 2 ( u 2 + u u 2 n) ( v 2 + v v 2 n) (b) If f(x) and g(x) are smooth realvalued functions, then f(x) g(x) d x Triangle Inequality For any vectors u and v, Proof We have 2 ( ) ( f(x) 2 d x u + v u + v g(x) 2 d x u + v 2 = u + v, u + v = u, u + u, v + v, u + v, v = u Re u, v + v 2 u 2 + u, v + v 2 u u v + v 2 = u 2 + v 2 )
4 Linear Algebra GramSchmidt Page 4 GramSchmidt Process It is often more simpler to work in an orthogonal basis GramSchmidt process is a method for obtaining orthonormal vectors from a linearly independent set of vectors in an inner product space, most commonly the Euclidean space IR n S = {v, v 2, v 3,, v k } and generates an orthogonal set S G = {u, u 2, u 3,, u k } or orthonormal basis S N = {e, e 2, e 3,, e k } that spans the same subspace of IR n as the set S v, u We define the projection operator by proj u (v) = u, where v, u denotes the inner u, u product of the vectors v and u This operator projects the vector v orthogonally onto the line spanned by vector u The GramSchmidt process then works as follows: u = v, e = u u u 2 = v 2 proj u (v 2 ), e 2 = u 2 u 2 u 3 = v 3 proj u (v 3 ) proj u2 (v 3 ), e 3 = u 3 u 3 u 4 = v 4 proj u (v 4 ) proj u2 (v 4 ) proj u3 (v 4 ), e 4 = u 4 u 4 = = k u k = v k proj uj (v k ), e k = u k u k j= The sequence {u, u 2,, u k } is the required system of orthogonal vectors, and the normalized vectors {e, e 2,, e k } form an orthonormal set The calculation of the sequence u, u 2,, u k is known as GramSchmidt orthogonalization, while the calculation of the sequence e, e 2,, e k is known as GramSchmidt orthonormalization as the vectors are normalized Example Consider the basis B = v =, v 2 =, v 3 = Then use Gram Schmidt to find the orthogonal basis B G = { u, u 2, u 3 } and the orthonormal basis B N = { e, e 2, e 3 } associated with B u = v = ; e = u u = 3 u 2 = v 2 v 2, u u, u u = = 2 2 = u 2 = 2 ; e 2 = u u 2 = 2 6 u 3 = v 3 v 3, u u, u u v 3, u 2 u 2, u 2 u 2 = + 2 = 0 ; e 3 = u u 3 = 0 2 Hence B G = u =, u 2 = 2, u 3 = 0 and B N = e =, e 2 = 2, e 3 =
5 Linear Algebra Matrix Norms Page 5 Matrix Norms A matrix norm is an extension of the notion of a vector norm to matrices It is a vector norm on IF m n That is, if A denotes the norm of the matrix A, then, A 0; 2 A = 0 if and only if A = 0; 3 α A = α A for all α IF and A IF m n ; 4 A, B IF m n, A + B A + B Additionally, in the case of square matrices (thus, m = n), some (but not all) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors: 5 AB A B for all matrices A and B in IF m n A matrix norm that satisfies this additional property is called a submultiplicative norm Operator Norm If vector norms on IF m and IF n are given (IF is the field of real or complex numbers), then one defines the corresponding operator norm or induced norm or, on the space of mbyn matrices as the following maxima: A = sup{ Ax : x IF n, x = } { } Ax = sup x : x IF n, x θ The operator norm corresponding to the pnorm for vectors is: Ax p A p = sup x 0 x p Note All operator norms are submultiplicative Here are some example matrix norms: m Column norm or norm: A = max j n i= a ij, which is simply the maximum absolute column sum of the matrix n Row norm or infinitynorm: A = max i m j= a ij, which is simply the maximum absolute row sum of the matrix The 2norm induced by the euclidean vector norm is A = λ max, where λ max is the largest eigenvalue of the matrix A A Although the Frobenius norm, defined as: k n A F = a ij 2 = trace (A A) i= j=
6 Linear Algebra Matrix Norms Page 6 is not an induced norm but it is a submultiplicative norm In MatLab, the Frobenius nom is invoked by the statement norm (A, fro ) The max norm A max = max a ij is neither an induced norm nor a submultiplicative norm The largest eigenvalue of a square matrix A in modulus, denoted by ρ (A) and called the spectral radius of A, never exceeds its operator norm: ρ (A) A p Given two normed vector spaces V and W (over the same base field, either the real numbers or the complex numbers), a linear map A : V W is continuous if and only if there exists a real number λ such that for all v V, Condition Number A v λ v The condition number associated with the linear equation A x = b, gives a bound on how inaccurate the solution x will be after approximation Note that this is before the effects of roundoff error are taken into account; conditioning is a property of the matrix, not the algorithm or floating point accuracy of the computer used to solve the corresponding system Let x 0 be the solution of the linear equation A x = b and let x be the calculated solution of A x = b, so the error in the solution is e = x x 0, and the relative error in the solution is e = x x 0 x 0 Generally, we can t verify the error or relative error (we don t know x 0 ), so a possible way to check this is by testing the accuracy back in the system A x = b : and relative residual is = residual r = A x 0 A x = b b, A x 0 A x A x = b b b = r b A matrix A is said to be illconditioned, if relatively small changes in the input (in the matrix A) can cause large change in the output (the solution of A x = b), ie; the solution is not very accurate if input is rounded Otherwise it is wellconditioned The condition number of an invertible matrix A is defined to be κ (A) = A A This quantity is always greater than or equal to The condition number is defined more precisely to be the maximum ratio of the relative error in x divided by the relative error in b If A is nonsingular, then by knowing the eigenvalues of A A (λ λ 2 λ n > 0), we can compute the condition number using the 2norm: κ 2 (A) = A 2 A 2 = λ λ n
7 Linear Algebra Matrix Norms Page 7 The smallest nonzero eigenvalue λ n of A A is a measure of how close the matrix is to being singular If λ n is small, then κ 2 (A) is large Thus, the closer the matrix is to being singular, the more illconditioned it is Consistent Norms A matrix norm ab on IF m n is called consistent with a vector norm a on IF n and a vector norm b on IF m, if: Ax b A ab u a, for all A IF m n, u IF n All induced norms are consistent by definition Compatible Norms A matrix norm b on IF n n is called compatible with a vector norm a on IF n if: A x a A b x a, for all A IF n n, x IF n All induced norms are compatible by definition Equivalence of Norms Two matrix norms α and β are said to be equivalent, if for all matrices A IF m n, there exist some positive numbers r and s such that Examples of Norm Equivalence r A α A β s A α For matrix A R \ of rank r, the following inequalities hold: A 2 A F r A 2 A F A r A F A max A 2 mn A max A A 2 m A n A A 2 n A m Here, A p refers to the matrix norm induced by the vector pnorm Another useful inequality between matrix norms is: A 2 A A
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