Orthogonal Definition and 583 Threads

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.

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  1. B

    Orthogonal Subgroups : on Modules?

    Hi, All: I have seen Orthogonal groups defined in relation to a pair (V,q) , where V is a vector space , and q is a symmetric, bilinear quadratic form. The orthogonal group associated with (V,q) is then the subgroup of GL(V) (invertible linear maps L:V-->V ), i.e., invertible matrices...
  2. S

    Orthogonal Matricies and rotations.

    Hi all, I have been trying to gain a deeper insight into quadratic forms and have realized that my textbook makes the assumption that an orthogonal matrix corresponds to either a rotation and/or reflection when viewed as a linear transformation. The textbook outlines a proof that demonstrates...
  3. Z

    Inverse problem for Orthogonal POlynomials

    given a set of orthogonal polynomials \int_{-\infty}^{\infty}dx P_{m} (x) P_{n} (x) w(x) = \delta _{m,n} the measure is EVEN and positive, so all the polynomials will be even or odd my question is if we suppose that for n-->oo \frac{ P_{2n} (x)}{P_{2n}(0)}= f(x) for a known...
  4. Y

    In a book I'm reading, it says:If beta is orthogonal to the set A,

    In a book I'm reading, it says: If beta is orthogonal to the set A, then beta is orthogonal to the closure of the linear span of A It's easy to see beta is orthogonal to the linear span of A, but I don't understand why it has to mention closure here?
  5. V

    Groups and orthogonal matrices question

    Let A and B be nxn matrices which generate a group under matrix multiplication. Assume A and B are not orthogonal. How can I determine an nxn matrix X such that X-1AX and X-1BX are both orthogonal matrices? Is it possible to do this without any special knowledge of the group in question?
  6. B

    Relationship Between Symplectic Group and Orthogonal Group

    Hi, All: Given a simplectic vector space (V,w), i.e., V is an n-dim. Vector Space ( n finite) and w is a symplectic form, i.e., a bilinear, antisymmetric totally isotropic and non-degenerate form, the simplectic groupSp(2n) of V is the (sub)group of GL(V) that...
  7. S

    Extend Vector to Orthogonal Basis

    How do you extend a vector (let's use vector (1,2,3) for example) to an orthogonal basis for R^3?
  8. M

    If U is an orthogonal matrix,its determinant is equal to 1 or -1.

    Question: Prove that is U is an orthogonal matrix, then the determinant of U is equal to 1 or -1. Hint consider the equation U^t = U^-1 and use the properties of the determinant. ------------------------------------------------------------------------------------------- So far I only...
  9. T

    Find the Orthogonal Complement

    Homework Statement Find the orthogonal complement. Homework Equations span {[0 0 1 0] [2 -1 0 1]} transpose The Attempt at a Solution I'm really confused on this. I've tried setting up things like this: [0 0 1 0] [2 -1 0 1] [x y z v] transpose = [0 0 0 0] transpose But...
  10. M

    Finding an Orthogonal Polynomial to x^2-1/2 on L2[0,1]

    Find a polynomial that is orthogonal to f(x)=x2-1/2 using L2[0,1]. I have looked all in the textbook and all over the internet and have found some hints if the interval is [-1,1], but still do not even know where to start here. I think I was gone the day our professor taught this because I do...
  11. Z

    Linear algebra Orthogonal Projections

    Homework Statement Given T is a projection such that ||Tx||≤||x||, prove T is an orthogonal projection. Homework Equations T:V\to V (V finite dimensional) <Tx,y>=<x,T^* y> general projection/idempotent operator: V=R(T)\oplus N(T) T^2=T orthogonal projection: R(T)=N(T)^{\perp}...
  12. P

    Proving vectors are orthogonal

    Homework Statement first the question asks find the jacobian matrix of (ucosv) (usinv ) ( w ) i have the matrix ( cos(v) , -usin(v) , 0) ( sin(v) , ucos(v) , 0) ( 0 , 0 , 1) the question asks to show that the columns are orthogonal...
  13. M

    Matrix of orthogonal projection

    Homework Statement Let A be the matrix of an orthogonal projection. Find A^2 in two ways: a. Geometrically. (consider what happens when you apply an orthogonal projection twice) b. By computation, using the formula: matrix of orthogonal projection onto V = QQ^T, where Q = [u1 ... um]...
  14. A

    How to Determine Orthogonal Vectors

    Hey guys, I have searched all over the forum but each thread seems to have a different way of solving this problem. I have changed the values from the coursework question so I can work it out for myself so here is an example one, I hope someone can give me some advice/steps on...
  15. C

    Orthogonal Complement in Inner Product Space: W2^\bot\subseteqW1^\bot

    Let W1 and W2 be subspaces of an inner product space V with W1\subseteqW2. Show that (the orthogonal complement denoted by \bot) W2^\bot\subseteqW1^\bot.
  16. K

    Find an orthogonal quantum state: introduction to dirac notation.

    Homework Statement Suppose we have a spin 1/2 Particle in a prepared state: \left|\Psi\right\rangle = \alpha \left|\uparrow\right\rangle + \beta\left|\downarrow\right\rangle where \left|\uparrow\right\rangle \left|\downarrow\right\rangle are orthonormal staes representing spin up and...
  17. P

    Really stuck computing orthogonal complement?

    Really stuck... computing orthogonal complement? Homework Statement The Attempt at a Solution :cry: I'm really sorry I can't provide much here because I really don't know how to proceed. Could anyone offer a hint to get me started?
  18. jegues

    Orthogonal wrt to linear polynomial(s)

    Homework Statement See figure attached for problem statement as well as my attempt. Homework Equations The Attempt at a Solution I can't see how we are expected to solve for 2 unknowns with only one equation? What am I missing? Am I supposed to simply define a in terms of b...
  19. Z

    Angular Velocity from Orthogonal Rotation Matrix

    Hi All, I have a rigidbody simulation and I'm trying to calculate the local angular velocity of the object using the derivative of it's orthogonal rotation matrix. This is where I'm stuck as I haven't been able to find an example on calculating the time derivative from two matrices at t=n and...
  20. T

    Transpose of orthogonal matrix

    Homework Statement Orthogonal matrix means Q^{T}Q=I, but not necessary QQ^{T}=I, so why can we say the inverse of Q is Q^{T}? Homework Equations The Attempt at a Solution the attempt is actually in my question. It's something i don't understand when doing revision.
  21. T

    Orthogonal complement (linear algebra)

    1. The problem statement let \vec x and \vec y be linearly independent vectors in R^n and let S=\text{span}(\vect x, \vect y). Define the matrix A as A=\vec x \vec y^T + \vec y \vec x^T. Show that N(A)=S^{\bot}. 2.equations I have a theorem that says N(A) = R(A^T)^{\bot}. A is symmetric; A...
  22. C

    Find vector orthogonal to two lines

    Homework Statement L1 pass through the points (-2,36,9) and (-8,44,12) L2 pass through the points (55,-31,7) and (41,-16,13) Find a point P on L1 and a point Q on L2 so that the vector \vec{}PQ is orthgonal to both lines. Homework Equations Dot product/ Cross product The Attempt at a...
  23. J

    Moment of inertia of hollow cylinder, axis orthogonal to length

    Hi, I am working through the Feynman lectures on physics and trying to calculate the moment of inertia stated in the title. (the taxis of rotation going through c.m., orthogonal to length). My approach is to slice the cylinder into thin rods along the length, using the parallel taxis theorem...
  24. G

    Construct & Normalize Orthogonal Vector to \underline{a}, \underline{b}

    Homework Statement Construct a third vector which is orthogonal to the following pair and normalize all three vectors: \underline{a}=(1-i,1,3i), \underline{b}=(1+2i,2,1) Homework Equations \underline{c}.\underline{a}=0 and \underline{c}.\underline{b}=0 where c=(x y z) The Attempt...
  25. Z

    Orthogonal Trajectories of Circles: Solving with Differentiation Method

    Homework Statement Find the orthogonal trajectories of the circles x2 + y2 - ay = 0The Attempt at a Solution I differentiated the equation w.r.t. x., Replaced dy/dx with -dx/dy, Solved the equation and got xyC = y2 - x2, where C is a constant. I did not eliminate 'a' after differentiating...
  26. G

    Diff Eq's - orthogonal polynomials

    Diff Eq's -- orthogonal polynomials [PLAIN]http://img27.imageshack.us/img27/566/39985815.jpg I managed to do the first part, stuck in the part circled. Any help will be appreciated, thanks.
  27. G

    Finding the Orthonormal Basis for a Given Subspace W

    Homework Statement Find the orthogonal projection of the given vector on the given subspace W of the inner product space V? V=R3, u = (2,1,3), and W = {(x,y,z): x + 3y - 2z = 0} I don't understand how to find the orthonormal basis for W? Homework Equations I don't understand how to...
  28. W

    Linear Algebra (eigenvectors, eigenvalues, and orthogonal projections)

    Homework Statement I am part way done with this problem... I don't know how to solve part e or part f. Any help or clues would be greatly appreciated. I have been trying to figure this out for a couple days now. W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it. a)...
  29. T

    Find Basis for Uperp from x1, x2, x3: Orthogonal Bases

    Homework Statement Let U be a subspace of R4 and let S={x1, x2, x3} be an orthogonal basis of U. Given x1, x2, x3, find a basis for Uperp (the subspace containing all vectors orthogonal to all vectors in U). I am actually given three vectors x1, x2, x3, but I am looking more to...
  30. A

    Orthogonal Matrix: Properties &amp; Conditions

    Dear all, I have a matrix, namely A. I calculate its eigenvalues by MATLAB and all of its eigenvalues lie on the unit circle(their amplitudes equal to 1). But A is not an orthogonal matrix (transpose(A) is not equal to inverse(A) ). What other condition or relationship may be correct for it?
  31. M

    Orthogonal functions with respect to a weight

    Homework Statement Say functions f and g continuous on [a,b] and happen to be orthogonal with respect to the weight function 1. Show that f or g has to vanish within (a,b). Homework Equations f and g are orthogonal w.r.t. a weight function w(x) if the integral from [a,b] of f(x)g(x)w(x)dx = 0...
  32. Z

    Solving Orthogonal Circles Problem

    Homework Statement A member of the family of the circles that cuts all the members of the family of circles x^2 + y^2 + 2gx + c=0 orthogonally, where c is a constant and g is a parameter is? Homework Equations The Attempt at a Solution Let the equation of the required circle...
  33. W

    Prove that the interior of the set of all orthogonal vectors to a is empty.

    Homework Statement Here is a picture of the problem: http://img84.imageshack.us/img84/1845/screenshot20100927at111.png If the link does not work, the problem basically asks: Let "a" be a non-zero vector in R^n. Let S be the set of all orthogonal vectors to "a" in R^n. I.e., a•x = 0 (where •...
  34. J

    Modelling a vacuum pipe using an orthogonal material

    Hi! I'm trying to model a vacuum pipe on Ansys. I've already modeled it using an isotropic material, assuming its a plane strain problem (infinate pipe), but now I want to use a laminate structure composed of an orthogonal material such as nomex (honeycomb) with a seal tight layer of Al or...
  35. 94JZA80

    Calc I problem regarding orthogonal curves and families of curves

    First, let me start by apologizing for the length of this post. i do in fact have a question about a problem that i couldn't solve (at least not the way i wanted to solve it), but first there is a fair amount of foreground information i must lay down...beyond this foreground information, this...
  36. T

    Show B|A| + A|B| and A|B| - B|A| are orthogonal.

    Homework Statement Show that B|A| + A|B| and A|B| - B|A| are orthogonal. Homework Equations Orthogonal meaning at right angles
  37. Jonnyb42

    Limit of orthogonal lines to straight line help?

    Yesterday I thought of a math problem, and it seems very simple, as I assume the solution is, and I want to know the answer more than I want to figure it out myself. Ok imagine point A and point B. The shortest path from A to B is a straight line. Let's now go from A to B in two orthogonal...
  38. F

    How to find a vector orthogonal to a line

    Hi I have two (two dimentional) linear equations (in the plane). and i am required to find out the angle between them. I have found the solution somewhere. the solution uses two vectors orthogal to the two lines. The problem now i have is: How to determine those orthogonal vectors? e.g: two...
  39. K

    Orthogonal Projection onto Hilbert Subspace

    Homework Statement I have a fixed unitary matrix, say X_d \in\mathfrak U(N) and a skew Hermitian matrix H \in \mathfrak u(N) . Consider the trace-inner product [tex] \langle A,B \rangle = \text{Tr}[A^\dagger B ] [/itex] where the dagger is the Hermitian transpose. I'm trying to find the...
  40. M

    Orthogonal diretion on the Minkowski diagram

    I have a trivial question: Let assume a world sheet of a time-like spherical shell in Minkowski space-time. On the 2D-Minkowski diagram (R,T), where R is the radius and T is the time, the world line is represented by a time-like curve. Let assume that the shell collapse and its 4-velocity is...
  41. D

    Orthogonal Projection Onto a Subspace?

    Hey, I have a linear algebra exam tomorrow and am finding it hard to figure out how to calculate an orthogonal projection onto a subspace. Here is the actual question type i am stuck on: I have spent ages searching the depths of google and other such places for a solution but with no...
  42. S

    Linear algebra geometry of orthogonal decomposition

    Homework Statement y=[4 8 1]^T u_1 = [2/3 1/3 2/3]^T u_2=[-2/3 2/3 1/3]^T Part 1: Write y as the sum of a vector y hat in W and a vector z in W complement Part 2: Describe the geometric relationship between the plane W in R^3 and the vectors y hat and z from the part above...
  43. K

    Orthogonal Complement to the Kernel of a Linear Transformation

    Hey all, I'm trying to find an orthogonal complement (under the standard inner product) to a space, and I think I've found the result mathematically. Unfortunately, when I apply the result to a toy example it seems to fail. Assume that A \in M_{m\times n}(\mathbb R^n), y \in \mathbb R^n and...
  44. Y

    Show that A is an orthogonal matrix

    Homework Statement If {aj} and {bj} are two separate sets of orthonormal basis sets, and are related by ai = \sumjnAijbj Show that A is an orthogonal matrix Homework Equations Provided above. The Attempt at a Solution Too much latex needed to show what I tried...
  45. N

    Two vectors u,v ∈ V are said to be orthogonal if

    Two vectors u,v ∈ V are said to be orthogonal if <u,v> = 0. Given the following statement: Two vectors u,v ∈ V are said to be orthogonal if <u,v> = 0. Is it correct to write it as: if <u,v> = 0, then the two vectors u,v ∈ V are said to be orthogonal or Is it correct to write it as...
  46. K

    Linear Algebra: Orthogonal basis ERG HELP

    Homework Statement Consider the vector V= [1 2 3 4]' in R4, find a basis of the subspace of R4 consisting of all vectors perpendicular to V. Homework Equations I mean, I'm just completely stumped by this one. I know that in R2, any V can be broken down to VParallel + VPerp, which...
  47. B

    2 screws orthogonal to each other -> compression

    Hey, I just fnished med school and I have been conduction biomechanical research for the past 3 years now. I am currently working on a project plan and I am encountering the following problem: We would like to measure compression across a joint, using partially threaded screws. For the...
  48. L

    Orthogonal transformation problem

    Homework Statement Lets say I fix 3 mutually orthogonal unit vectors i, j and k. Consider an orthogonal transformation F of vectors defined by F(a_1i+ a_2j + a_3k)=a_1'i+a_2'k+a_3'k where \left( \begin{array}{ccc} a_1 \\ a_2 \\ a_3\end{array}\right) = A\left( \begin{array}{ccc} a_1' \\...
  49. M

    Question on orthogonal eigenfunctions

    in this book I have by G.L Squires. One of the questions is: if \phi1 and \phi2 are normalized eigenfunctions corresponding to the same eigenvalue. If: \int\phi1*\phi2 d\tau = d where d is real, find normalized linear combinations of \phi1 and \phi 2 that are orthogonal to a) \phi 1 b)...
  50. mnb96

    Orthogonal and symmetric matrices

    Hello, I guess this is a basic question. Let´s say that If I am given a matrix X it is possible to form a symmetric matrix by computing X+X^{T} . But how can I form a matrix which is both symmetric and orthogonal? That is: M=M^{T}=M^{-1}.
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