Week 8-9. Inner Product Spaces. (Revised Version) Section 3.1 Dot Product As an Inner Product
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21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Partitioned (Or Block) Matrices This Version: 29 Nov 2018
Partitioned (or Block) Matrices This version: 29 Nov 2018 Intermediate Econometrics / Forecasting Class Notes Instructor: Anthony Tay It is frequently convenient to partition matrices into smaller sub-matrices. e.g. 2 3 2 1 3 2 3 2 1 3 4 1 1 0 7 4 1 1 0 7 A B (2×2) (2×3) 3 1 1 0 0 = 3 1 1 0 0 = C I 1 3 0 1 0 1 3 0 1 0 (3×2) (3×3) 2 0 0 0 1 2 0 0 0 1 The same matrix can be partitioned in several different ways. For instance, we can write the previous matrix as 2 3 2 1 3 2 3 2 1 3 4 1 1 0 7 4 1 1 0 7 a b0 (1×1) (1×4) 3 1 1 0 0 = 3 1 1 0 0 = c D 1 3 0 1 0 1 3 0 1 0 (4×1) (4×4) 2 0 0 0 1 2 0 0 0 1 One reason partitioning is useful is that we can do matrix addition and multiplication with blocks, as though the blocks are elements, as long as the blocks are conformable for the operations. For instance: A B D E A + D B + E (2×2) (2×3) (2×2) (2×3) (2×2) (2×3) + = C I C F 2C I + F (3×2) (3×3) (3×2) (3×3) (3×2) (3×3) A B d E Ad + BF AE + BG (2×2) (2×3) (2×1) (2×3) (2×1) (2×3) = C I F G Cd + F CE + G (3×2) (3×3) (3×1) (3×3) (3×1) (3×3) | {z } | {z } | {z } (5×5) (5×4) (5×4) 1 Intermediate Econometrics / Forecasting 2 Examples (1) Let 1 2 1 1 2 1 c 1 4 2 3 4 2 3 h i A = = = a a a and c = c 1 2 3 2 3 0 1 3 0 1 c 0 1 3 0 1 3 3 c1 h i then Ac = a1 a2 a3 c2 = c1a1 + c2a2 + c3a3 c3 The product Ac produces a linear combination of the columns of A. -
Tight Frames and Their Symmetries
Technical Report 9 December 2003 Tight Frames and their Symmetries Richard Vale, Shayne Waldron Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: [email protected] (http:www.math.auckland.ac.nz/˜waldron) e–mail: [email protected] ABSTRACT The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries. Key Words: Tight frames, isometric tight frames, Gram matrix, multivariate orthogonal polynomials, symmetry groups, harmonic frames, representation theory, wavelets AMS (MOS) Subject Classifications: primary 05B20, 33C50, 20C15, 42C15, sec- ondary 52B15, 42C40 0 1. Introduction u1 u2 u3 2 The three equally spaced unit vectors u1, u2, u3 in IR provide the following redundant representation 2 3 f = f, u u , f IR2, (1.1) 3 h ji j ∀ ∈ j=1 X which is the simplest example of a tight frame. Such representations arose in the study of nonharmonic Fourier series in L2(IR) (see Duffin and Schaeffer [DS52]) and have recently been used extensively in the theory of wavelets (see, e.g., Daubechies [D92]). -
Handout 9 More Matrix Properties; the Transpose
Handout 9 More matrix properties; the transpose Square matrix properties These properties only apply to a square matrix, i.e. n £ n. ² The leading diagonal is the diagonal line consisting of the entries a11, a22, a33, . ann. ² A diagonal matrix has zeros everywhere except the leading diagonal. ² The identity matrix I has zeros o® the leading diagonal, and 1 for each entry on the diagonal. It is a special case of a diagonal matrix, and A I = I A = A for any n £ n matrix A. ² An upper triangular matrix has all its non-zero entries on or above the leading diagonal. ² A lower triangular matrix has all its non-zero entries on or below the leading diagonal. ² A symmetric matrix has the same entries below and above the diagonal: aij = aji for any values of i and j between 1 and n. ² An antisymmetric or skew-symmetric matrix has the opposite entries below and above the diagonal: aij = ¡aji for any values of i and j between 1 and n. This automatically means the digaonal entries must all be zero. Transpose To transpose a matrix, we reect it across the line given by the leading diagonal a11, a22 etc. In general the result is a di®erent shape to the original matrix: a11 a21 a11 a12 a13 > > A = A = 0 a12 a22 1 [A ]ij = A : µ a21 a22 a23 ¶ ji a13 a23 @ A > ² If A is m £ n then A is n £ m. > ² The transpose of a symmetric matrix is itself: A = A (recalling that only square matrices can be symmetric). -
Gram Matrix and Orthogonality in Frames 1
U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 1, 2018 ISSN 1223-7027 GRAM MATRIX AND ORTHOGONALITY IN FRAMES Abolhassan FEREYDOONI1 and Elnaz OSGOOEI 2 In this paper, we aim at introducing a criterion that determines if f figi2I is a Bessel sequence, a frame or a Riesz sequence or not any of these, based on the norms and the inner products of the elements in f figi2I. In the cases of Riesz and Bessel sequences, we introduced a criterion but in the case of a frame, we did not find any answers. This criterion will be shown by K(f figi2I). Using the criterion introduced, some interesting extensions of orthogonality will be presented. Keywords: Frames, Bessel sequences, Orthonormal basis, Riesz bases, Gram matrix MSC2010: Primary 42C15, 47A05. 1. Preliminaries Frames are generalizations of orthonormal bases, but, more than orthonormal bases, they have shown their ability and stability in the representation of functions [1, 4, 10, 11]. The frames have been deeply studied from an abstract point of view. The results of such studies have been used in concrete frames such as Gabor and Wavelet frames which are very important from a practical point of view [2, 9, 5, 8]. An orthonormal set feng in a Hilbert space is characterized by a simple relation hem;eni = dm;n: In the other words, the Gram matrix is the identity matrix. Moreover, feng is an orthonor- mal basis if spanfeng = H. But, for frames the situation is more complicated; i.e., the Gram Matrix has no such a simple form. -
Uniqueness of Low-Rank Matrix Completion by Rigidity Theory
UNIQUENESS OF LOW-RANK MATRIX COMPLETION BY RIGIDITY THEORY AMIT SINGER∗ AND MIHAI CUCURINGU† Abstract. The problem of completing a low-rank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and control. Most recent work had been focused on constructing efficient algorithms for exact or approximate recovery of the missing matrix entries and proving lower bounds for the number of known entries that guarantee a successful recovery with high probability. A related problem from both the mathematical and algorithmic point of view is the distance geometry problem of realizing points in a Euclidean space from a given subset of their pairwise distances. Rigidity theory answers basic questions regarding the uniqueness of the realization satisfying a given partial set of distances. We observe that basic ideas and tools of rigidity theory can be adapted to determine uniqueness of low-rank matrix completion, where inner products play the role that distances play in rigidity theory. This observation leads to efficient randomized algorithms for testing necessary and sufficient conditions for local completion and for testing sufficient conditions for global completion. Crucial to our analysis is a new matrix, which we call the completion matrix, that serves as the analogue of the rigidity matrix. Key words. Low rank matrices, missing values, rigidity theory, iterative methods, collaborative filtering. AMS subject classifications. 05C10, 05C75, 15A48 1. Introduction. Can the missing entries of an incomplete real valued matrix be recovered? Clearly, a matrix can be completed in an infinite number of ways by replacing the missing entries with arbitrary values. -
New Foundations for Geometric Algebra1
Text published in the electronic journal Clifford Analysis, Clifford Algebras and their Applications vol. 2, No. 3 (2013) pp. 193-211 New foundations for geometric algebra1 Ramon González Calvet Institut Pere Calders, Campus Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain E-mail : [email protected] Abstract. New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multiple quantities. -
Matrix Determinants
MATRIX DETERMINANTS Summary Uses ................................................................................................................................................. 1 1‐ Reminder ‐ Definition and components of a matrix ................................................................ 1 2‐ The matrix determinant .......................................................................................................... 2 3‐ Calculation of the determinant for a matrix ................................................................. 2 4‐ Exercise .................................................................................................................................... 3 5‐ Definition of a minor ............................................................................................................... 3 6‐ Definition of a cofactor ............................................................................................................ 4 7‐ Cofactor expansion – a method to calculate the determinant ............................................... 4 8‐ Calculate the determinant for a matrix ........................................................................ 5 9‐ Alternative method to calculate determinants ....................................................................... 6 10‐ Exercise .................................................................................................................................... 7 11‐ Determinants of square matrices of dimensions 4x4 and greater ........................................ -
Inner Products and Norms (Part III)
Inner Products and Norms (part III) Prof. Dan A. Simovici UMB 1 / 74 Outline 1 Approximating Subspaces 2 Gram Matrices 3 The Gram-Schmidt Orthogonalization Algorithm 4 QR Decomposition of Matrices 5 Gram-Schmidt Algorithm in R 2 / 74 Approximating Subspaces Definition A subspace T of a inner product linear space is an approximating subspace if for every x 2 L there is a unique element in T that is closest to x. Theorem Let T be a subspace in the inner product space L. If x 2 L and t 2 T , then x − t 2 T ? if and only if t is the unique element of T closest to x. 3 / 74 Approximating Subspaces Proof Suppose that x − t 2 T ?. Then, for any u 2 T we have k x − u k2=k (x − t) + (t − u) k2=k x − t k2 + k t − u k2; by observing that x − t 2 T ? and t − u 2 T and applying Pythagora's 2 2 Theorem to x − t and t − u. Therefore, we have k x − u k >k x − t k , so t is the unique element of T closest to x. 4 / 74 Approximating Subspaces Proof (cont'd) Conversely, suppose that t is the unique element of T closest to x and x − t 62 T ?, that is, there exists u 2 T such that (x − t; u) 6= 0. This implies, of course, that u 6= 0L. We have k x − (t + au) k2=k x − t − au k2=k x − t k2 −2(x − t; au) + jaj2 k u k2 : 2 2 Since k x − (t + au) k >k x − t k (by the definition of t), we have 2 2 1 −2(x − t; au) + jaj k u k > 0 for every a 2 F. -
Exceptional Collections in Surface-Like Categories 11
EXCEPTIONAL COLLECTIONS IN SURFACE-LIKE CATEGORIES ALEXANDER KUZNETSOV Abstract. We provide a categorical framework for recent results of Markus Perling on combinatorics of exceptional collections on numerically rational surfaces. Using it we simplify and generalize some of Perling’s results as well as Vial’s criterion for existence of a numerical exceptional collection. 1. Introduction A beautiful recent paper of Markus Perling [Pe] proves that any numerically exceptional collection of maximal length in the derived category of a numerically rational surface (i.e., a surface with zero irreg- ularity and geometric genus) can be transformed by mutations into an exceptional collection consisting of objects of rank 1. In this note we provide a categorical framework that allows to extend and simplify Perling’s result. For this we introduce a notion of a surface-like category. Roughly speaking, it is a triangulated cate- T num T gory whose numerical Grothendieck group K0 ( ), considered as an abelian group with a bilinear form (Euler form), behaves similarly to the numerical Grothendieck group of a smooth projective sur- face, see Definition 3.1. Of course, for any smooth projective surface X its derived category D(X) is surface-like. However, there are surface-like categories of different nature, for instance derived categories of noncommutative surfaces and of even-dimensional Calabi–Yau varieties turn out to be surface-like (Example 3.4). Also, some subcategories of surface-like categories are surface-like. Thus, the notion of a surface-like category is indeed more general. In fact, all results of this paper have a numerical nature, so instead of considering categories, we pass directly to their numerical Grothendieck groups. -
6 Inner Product Spaces
Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space quite a special type of a vector space. Essentially all these are consequences of the dot product. Thus, it makes sense to look for operations which share the basic properties of the dot product. In this section we shall briefly discuss this. Definition 6.1 Let V be a vector space. By an inner product on V we mean a binary operation, which associates a scalar say u, v for each pair of vectors u, v) in V, satisfying h i the following properties for all u, v, w in V and α, β any scalar. (Let “ ” denote the complex − conjugate of a complex number.) (1) u, v = v, u (Hermitian property or conjugate symmetry); h i h i (2) u, αv + βw = α u, v + β u, w (sesquilinearity); h i h i h i (3) v, v > 0 if v =0 (positivity). h i 6 A vector space with an inner product is called an inner product space. Remark 6.1 (i) Observe that we have not mentioned whether V is a real vector space or a complex vector space. The above definition includes both the cases. The only difference is that if K = R then the conjugation is just the identity. Thus for real vector spaces, (1) will becomes ‘symmetric property’ since for a real number c, we have c¯ = c. (ii) Combining (1) and (2) we obtain (2’) αu + βv, w =α ¯ u, w + β¯ v, w . -
Applications of the Wronskian and Gram Matrices of {Fie”K?
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applications of the Wronskian and Gram Matrices of {fie”k? Robert E. Hartwig Mathematics Department North Carolina State University Raleigh, North Carolina 27650 Submitted by S. Barn&t ABSTRACT A review is made of some of the fundamental properties of the sequence of functions {t’b’}, k=l,..., s, i=O ,..., m,_,, with distinct X,. In particular it is shown how the Wronskian and Gram matrices of this sequence of functions appear naturally in such fields as spectral matrix theory, controllability, and Lyapunov stability theory. 1. INTRODUCTION Let #(A) = II;,,< A-hk)“‘k be a complex polynomial with distinct roots x 1,.. ., A,, and let m=ml+ . +m,. It is well known [9] that the functions {tieXk’}, k=l,..., s, i=O,l,..., mkpl, form a fundamental (i.e., linearly inde- pendent) solution set to the differential equation m)Y(t)=O> (1) where D = $(.). It is the purpose of this review to illustrate some of the important properties of this independent solution set. In particular we shall show that both the Wronskian matrix and the Gram matrix play a dominant role in certain applications of matrix theory, such as spectral theory, controllability, and the Lyapunov stability theory. Few of the results in this paper are new; however, the proofs we give are novel and shed some new light on how the various concepts are interrelated, and on why certain results work the way they do. LINEAR ALGEBRA ANDITSAPPLlCATIONS43:229-241(1982) 229 C Elsevier Science Publishing Co., Inc., 1982 52 Vanderbilt Ave., New York, NY 10017 0024.3795/82/020229 + 13$02.50 230 ROBERT E.