Orthogonality and Least Squares
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A New Description of Space and Time Using Clifford Multivectors
A new description of space and time using Clifford multivectors James M. Chappell† , Nicolangelo Iannella† , Azhar Iqbal† , Mark Chappell‡ , Derek Abbott† †School of Electrical and Electronic Engineering, University of Adelaide, South Australia 5005, Australia ‡Griffith Institute, Griffith University, Queensland 4122, Australia Abstract Following the development of the special theory of relativity in 1905, Minkowski pro- posed a unified space and time structure consisting of three space dimensions and one time dimension, with relativistic effects then being natural consequences of this space- time geometry. In this paper, we illustrate how Clifford’s geometric algebra that utilizes multivectors to represent spacetime, provides an elegant mathematical framework for the study of relativistic phenomena. We show, with several examples, how the application of geometric algebra leads to the correct relativistic description of the physical phenomena being considered. This approach not only provides a compact mathematical representa- tion to tackle such phenomena, but also suggests some novel insights into the nature of time. Keywords: Geometric algebra, Clifford space, Spacetime, Multivectors, Algebraic framework 1. Introduction The physical world, based on early investigations, was deemed to possess three inde- pendent freedoms of translation, referred to as the three dimensions of space. This naive conclusion is also supported by more sophisticated analysis such as the existence of only five regular polyhedra and the inverse square force laws. If we lived in a world with four spatial dimensions, for example, we would be able to construct six regular solids, and in arXiv:1205.5195v2 [math-ph] 11 Oct 2012 five dimensions and above we would find only three [1]. -
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. -
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). -
Lecture 4: April 8, 2021 1 Orthogonality and Orthonormality
Mathematical Toolkit Spring 2021 Lecture 4: April 8, 2021 Lecturer: Avrim Blum (notes based on notes from Madhur Tulsiani) 1 Orthogonality and orthonormality Definition 1.1 Two vectors u, v in an inner product space are said to be orthogonal if hu, vi = 0. A set of vectors S ⊆ V is said to consist of mutually orthogonal vectors if hu, vi = 0 for all u 6= v, u, v 2 S. A set of S ⊆ V is said to be orthonormal if hu, vi = 0 for all u 6= v, u, v 2 S and kuk = 1 for all u 2 S. Proposition 1.2 A set S ⊆ V n f0V g consisting of mutually orthogonal vectors is linearly inde- pendent. Proposition 1.3 (Gram-Schmidt orthogonalization) Given a finite set fv1,..., vng of linearly independent vectors, there exists a set of orthonormal vectors fw1,..., wng such that Span (fw1,..., wng) = Span (fv1,..., vng) . Proof: By induction. The case with one vector is trivial. Given the statement for k vectors and orthonormal fw1,..., wkg such that Span (fw1,..., wkg) = Span (fv1,..., vkg) , define k u + u = v − hw , v i · w and w = k 1 . k+1 k+1 ∑ i k+1 i k+1 k k i=1 uk+1 We can now check that the set fw1,..., wk+1g satisfies the required conditions. Unit length is clear, so let’s check orthogonality: k uk+1, wj = vk+1, wj − ∑ hwi, vk+1i · wi, wj = vk+1, wj − wj, vk+1 = 0. i=1 Corollary 1.4 Every finite dimensional inner product space has an orthonormal basis. -
Week 8-9. Inner Product Spaces. (Revised Version) Section 3.1 Dot Product As an Inner Product
Math 2051 W2008 Margo Kondratieva Week 8-9. Inner product spaces. (revised version) Section 3.1 Dot product as an inner product. Consider a linear (vector) space V . (Let us restrict ourselves to only real spaces that is we will not deal with complex numbers and vectors.) De¯nition 1. An inner product on V is a function which assigns a real number, denoted by < ~u;~v> to every pair of vectors ~u;~v 2 V such that (1) < ~u;~v>=< ~v; ~u> for all ~u;~v 2 V ; (2) < ~u + ~v; ~w>=< ~u;~w> + < ~v; ~w> for all ~u;~v; ~w 2 V ; (3) < k~u;~v>= k < ~u;~v> for any k 2 R and ~u;~v 2 V . (4) < ~v;~v>¸ 0 for all ~v 2 V , and < ~v;~v>= 0 only for ~v = ~0. De¯nition 2. Inner product space is a vector space equipped with an inner product. Pn It is straightforward to check that the dot product introduces by ~u ¢ ~v = j=1 ujvj is an inner product. You are advised to verify all the properties listed in the de¯nition, as an exercise. The dot product is also called Euclidian inner product. De¯nition 3. Euclidian vector space is Rn equipped with Euclidian inner product < ~u;~v>= ~u¢~v. De¯nition 4. A square matrix A is called positive de¯nite if ~vT A~v> 0 for any vector ~v 6= ~0. · ¸ 2 0 Problem 1. Show that is positive de¯nite. 0 3 Solution: Take ~v = (x; y)T . Then ~vT A~v = 2x2 + 3y2 > 0 for (x; y) 6= (0; 0). -
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 ........................................ -
Orthogonality Handout
3.8 (SUPPLEMENT) | ORTHOGONALITY OF EIGENFUNCTIONS We now develop some properties of eigenfunctions, to be used in Chapter 9 for Fourier Series and Partial Differential Equations. 1. Definition of Orthogonality R b We say functions f(x) and g(x) are orthogonal on a < x < b if a f(x)g(x) dx = 0 . [Motivation: Let's approximate the integral with a Riemann sum, as follows. Take a large integer N, put h = (b − a)=N and partition the interval a < x < b by defining x1 = a + h; x2 = a + 2h; : : : ; xN = a + Nh = b. Then Z b f(x)g(x) dx ≈ f(x1)g(x1)h + ··· + f(xN )g(xN )h a = (uN · vN )h where uN = (f(x1); : : : ; f(xN )) and vN = (g(x1); : : : ; g(xN )) are vectors containing the values of f and g. The vectors uN and vN are said to be orthogonal (or perpendicular) if their dot product equals zero (uN ·vN = 0), and so when we let N ! 1 in the above formula it makes R b sense to say the functions f and g are orthogonal when the integral a f(x)g(x) dx equals zero.] R π 1 2 π Example. sin x and cos x are orthogonal on −π < x < π, since −π sin x cos x dx = 2 sin x −π = 0. 2. Integration Lemma Suppose functions Xn(x) and Xm(x) satisfy the differential equations 00 Xn + λnXn = 0; a < x < b; 00 Xm + λmXm = 0; a < x < b; for some numbers λn; λm. Then Z b 0 0 b (λn − λm) Xn(x)Xm(x) dx = [Xn(x)Xm(x) − Xn(x)Xm(x)]a: a Proof. -
Inner Products and Orthogonality
Advanced Linear Algebra – Week 5 Inner Products and Orthogonality This week we will learn about: • Inner products (and the dot product again), • The norm induced by the inner product, • The Cauchy–Schwarz and triangle inequalities, and • Orthogonality. Extra reading and watching: • Sections 1.3.4 and 1.4.1 in the textbook • Lecture videos 17, 18, 19, 20, 21, and 22 on YouTube • Inner product space at Wikipedia • Cauchy–Schwarz inequality at Wikipedia • Gram–Schmidt process at Wikipedia Extra textbook problems: ? 1.3.3, 1.3.4, 1.4.1 ?? 1.3.9, 1.3.10, 1.3.12, 1.3.13, 1.4.2, 1.4.5(a,d) ??? 1.3.11, 1.3.14, 1.3.15, 1.3.25, 1.4.16 A 1.3.18 1 Advanced Linear Algebra – Week 5 2 There are many times when we would like to be able to talk about the angle between vectors in a vector space V, and in particular orthogonality of vectors, just like we did in Rn in the previous course. This requires us to have a generalization of the dot product to arbitrary vector spaces. Definition 5.1 — Inner Product Suppose that F = R or F = C, and V is a vector space over F. Then an inner product on V is a function h·, ·i : V × V → F such that the following three properties hold for all c ∈ F and all v, w, x ∈ V: a) hv, wi = hw, vi (conjugate symmetry) b) hv, w + cxi = hv, wi + chv, xi (linearity in 2nd entry) c) hv, vi ≥ 0, with equality if and only if v = 0. -
Eigenvalues, Eigenvectors and the Similarity Transformation
Eigenvalues, Eigenvectors and the Similarity Transformation Eigenvalues and the associated eigenvectors are ‘special’ properties of square matrices. While the eigenvalues parameterize the dynamical properties of the system (timescales, resonance properties, amplification factors, etc) the eigenvectors define the vector coordinates of the normal modes of the system. Each eigenvector is associated with a particular eigenvalue. The general state of the system can be expressed as a linear combination of eigenvectors. The beauty of eigenvectors is that (for square symmetric matrices) they can be made orthogonal (decoupled from one another). The normal modes can be handled independently and an orthogonal expansion of the system is possible. The decoupling is also apparent in the ability of the eigenvectors to diagonalize the original matrix, A, with the eigenvalues lying on the diagonal of the new matrix, . In analogy to the inertia tensor in mechanics, the eigenvectors form the principle axes of the solid object and a similarity transformation rotates the coordinate system into alignment with the principle axes. Motion along the principle axes is decoupled. The matrix mechanics is closely related to the more general singular value decomposition. We will use the basis sets of orthogonal eigenvectors generated by SVD for orbit control problems. Here we develop eigenvector theory since it is more familiar to most readers. Square matrices have an eigenvalue/eigenvector equation with solutions that are the eigenvectors xand the associated eigenvalues : Ax = x The special property of an eigenvector is that it transforms into a scaled version of itself under the operation of A. Note that the eigenvector equation is non-linear in both the eigenvalue () and the eigenvector (x). -
A Guided Tour to the Plane-Based Geometric Algebra PGA
A Guided Tour to the Plane-Based Geometric Algebra PGA Leo Dorst University of Amsterdam Version 1.15{ July 6, 2020 Planes are the primitive elements for the constructions of objects and oper- ators in Euclidean geometry. Triangulated meshes are built from them, and reflections in multiple planes are a mathematically pure way to construct Euclidean motions. A geometric algebra based on planes is therefore a natural choice to unify objects and operators for Euclidean geometry. The usual claims of `com- pleteness' of the GA approach leads us to hope that it might contain, in a single framework, all representations ever designed for Euclidean geometry - including normal vectors, directions as points at infinity, Pl¨ucker coordinates for lines, quaternions as 3D rotations around the origin, and dual quaternions for rigid body motions; and even spinors. This text provides a guided tour to this algebra of planes PGA. It indeed shows how all such computationally efficient methods are incorporated and related. We will see how the PGA elements naturally group into blocks of four coordinates in an implementation, and how this more complete under- standing of the embedding suggests some handy choices to avoid extraneous computations. In the unified PGA framework, one never switches between efficient representations for subtasks, and this obviously saves any time spent on data conversions. Relative to other treatments of PGA, this text is rather light on the mathematics. Where you see careful derivations, they involve the aspects of orientation and magnitude. These features have been neglected by authors focussing on the mathematical beauty of the projective nature of the algebra.