What is Orthogonal: Definition and 581 Discussions

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. M

    Gram Schmidt Method (calculating orthogonal vector set)

    Hey, I'm going over the Gram Schmidt method, and need some help understanding it. I understand that you're intending to create an orthogonal vector (i'll call them v) set based on a set of vectors you already have (i'll call them u). Then: Let v1 = u1 Now, construct the second orthogonal...
  2. C

    Orthogonal Matrix - Linear Algebra

    Homework Statement [PLAIN]http://img504.imageshack.us/img504/4985/capturewm.jpg Homework Equations N/A The Attempt at a Solution This is more of a conceptual question so I need a little help knowing what kinds of things to look for.
  3. D

    Prove Orthogonal Vectors: x ⊥ u and v implies x ⊥ u - v

    If x is ⊥ u and v, then x is ⊥ u - v. I know this is true because u - v is in the same place as u and v; therefore, x is orthogonal. How can this be written better?
  4. G

    Basis of Orthogonal Complement

    Let S be the subspace of R^3 spanned by x=(1,-1,1)^T. Find a basis for the orthogonal complement of S. I don't even know where to start... I would appreciate your help!
  5. R

    Linear independence of orthogonal and orthonormal sets?

    (Note: this isn't a homework question, I'm reviewing and I think the textbook is wrong.) I'm working through the Gram-Schmidt process in my textbook, and at the end of every chapter it starts the problem set with a series of true or false questions. One statement is: -Every orthogonal set...
  6. G

    Transitive Property with Orthogonal Vectors?

    Homework Statement Let x1, x2, and x3 be vectors in R^3. If x1 is orthogonal to x2 and x2 is orthogonal to x3, is it necessarily true that x1 is orthogonal to x3? Homework Equations I know that if x1 is orthogonal to x2 and x2 is orthogonal to x3, then... (x1)^T*x2=0 (x2)^T*x3=0...
  7. T

    How to prove that an orthogonal projection matrix is idempotent

    Homework Statement Prove that [P]^2=[P] (that the matrix is idempotent) Homework Equations The Attempt at a Solution A(A^T*A)^-1 A^T= (A(A^T*A)^-1 A^T)^2 Where A^T is the transpose of A. I have no idea.
  8. G

    Find Orthogonal Matrix with 1st Row (1/3,2/3,2/3)

    Find an orthogonal matrix whose first row is (1/3,2/3,2/3) I know orthogonal matrix A satisfies A*A' = I, where A' is the transpose of A and I is identity matrix. Let A = 1/3*{{1,2,3},{a,b,c},{d,e,f}} where a,b,c,d,e,f elements of R A'= 1/3*{{1,a,d},{2,b,e},{2,c,f}} We can obtain...
  9. S

    Linear algebra - Orthogonal matrix

    Homework Statement Let T: Rn -> Rn be a linear transformation, and let B be an orthonormal basis for R^n. Prove that [ the length of T(x) ] = [ the length of x ] if and only if [T]B (the B-matrix for T) is an orthogonal matrix. Homework Equations None I don't think. The Attempt at...
  10. R

    Linear Algebra- find an orthogonal matrix with eigenvalue=1 or -1

    Homework Statement I have to find an orthogonal matrix with an eigenvalue that does not equal 1 or -1. That's it. I'm completely stumped. Homework Equations An orthogonal matrix is defined as a matrix whose columns are an orthonormal basis, that is they are all orthogonal to each other...
  11. I

    Show eigenfunctions are orthogonal

    hi one of my past papers needs me to show that if 2 eigenfunctions, A and B, of an operator O possesses different eigenvalues, a and b, they must be orthogonal. assume eigenvalues are real. we are given \int A*OB dx = \int(OA)*B dx * indicates conjugate
  12. D

    Factor the matrix into the form QR where Q is orthogonal

    Factor the matrix into the form QR where Q is orthogonal and R is upper triangular. \begin{bmatrix} a & b\\ c & d \end{bmatrix}*\begin{bmatrix} e & f\\ 0 & g \end{bmatrix}=\begin{bmatrix} -1 & 3\\ 1 & 5 \end{bmatrix} \begin{bmatrix} a & c \end{bmatrix}*\begin{bmatrix} b\\...
  13. D

    Prove Orthogonal Matrix Transpose is Orthogonal

    Prove that the transpose of an orthogonal matrix is an orthogonal matrix. I think that the Kronecker delta needs to be used but not sure how to write it out.
  14. M

    Orthogonal Properties for Sine Don't Hold if Pi is involded?

    Orthogonal Properties for Sine Don't Hold if Pi is involded?? Normally I know \int_{-L}^L \sin \frac{n x}{L} \sin \frac{\m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m but apparently this doesn't work for \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx I am...
  15. X

    A plane wave vs the bound state of Hydrogen atom: orthogonal?

    These days I met one problem and asked a professor for help. But I can not understand his answer. Can you help me explain his answer? My question is that whether we can assume that a plane wave is orthogonal to the bound state of Hydrogen atom when t->\infty? Professor answers...
  16. Z

    Limit of orthogonal polynomials for big n

    given the 'normalized' Chebyshev and Legendre Polynomials \frac{L_{2n}(x)}{L_{2n}(0)} and \frac{T_{2n}(x)}{T_{2n}(0)} for n even and BIG 2n--->oo then it would be true that (in this limit) \frac{L_{2n}(x)}{L_{2n}(0)}=\frac{sin(x)}{2x} and \frac{T_{2n}(x)}{T_{2n}(0)}=J_{0}(2x) here...
  17. R

    How can I determine an orthogonal vector to a given vector in 3D space?

    Hey guys, Given a vector, ie < -1, 2, 3 > , how does one go about finding a vector which is orthogonal to it? I also have another vector < x, y ,z > which is the point of origin for the above vector. In context, I'm given a directional vector from which I need to find an 'up' vector and a...
  18. N

    How do I find two vectors that are orthogonal to each other?

    1. Find a nonzero vector v in span {v2,v3} such that v is orthogonal to v3. Express v as a linear combination of v2 and v3 2. v1= [3 5 11] v2= [5 9 20] v3= [11 20 49] 3. I know that the dot product of v and v3 must equal zero. And that v must have components between 5 and 11, 9 and...
  19. V

    What are the values of s that make two given vectors orthogonal?

    By evaluating the dot product, find the values of the scalar s for which the two vectors b=X+sY and c=X-sY are orthogonal also explain your answers with a sketch: my working (X,sY).(X,-sY) has to equal 0 for them to be orthogonal x.x = 1 since they are unit vectors...
  20. e2m2a

    Orthogonal angle with respect to all inertial frames?

    Here is a thought experiment that I cannot resolve. Maybe someone smarter than I can do this. Suppose we have a long, thin rod rotating in the counter clockwise direction. The rod rotates around an axis which is connected to one end of the rod. The axis is attached to a second object which...
  21. Z

    Orthogonal matrices prove: T is orthogonal iff [T]_bb is an orthogonal matrix

    Homework Statement Let B = {v1, ..., vn} be an arbitrary orthonormal basis of Rn, prove T is orthogonal iff [T]_{BB} is an orthogonal matrix. Hint: If B is orhtogonal basis for Rn then, x.y = [x]_B . [y]_Bfor all x, y in Rn. 3. The Attempt at a Solution If [T]_{BB} is an orthogonal matrix...
  22. T

    Bessel equation & Orthogonal Basis

    I remember some of my linear algebra from my studies but can't wrap my head around this one. Homework Statement Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So...
  23. F

    What is an Orthogonal Family of Curves?

    Homework Statement Anyone familiar with orthogonal families of curves? They're not that difficult to understand. If you have a differential equation \frac{dy}{dx} = F(x, y) you can find it's orthogonal family of curves by solving for \frac{dy}{dx} = \frac{-1}{F(x, y)} Homework...
  24. K

    Need help on Orthogonal Trajectories in my Diff. EQ. Class

    Homework Statement Show that the families (x+c1)(x2+y2)+x = 0 and (y+c2)(x2+y2)-y = 0 Homework Equations For the 2 curves to be orthogonal their slopes should be negative recriprocles. The Attempt at a Solution I'm pretty sure that for the first set of curves: y'(x) = - (2c1...
  25. R

    Is it vacum state is orthogonal or not?

    Homework Statement \left\langle0_{x}|0_{y}\right\rangle is this orthogonal or not? Homework Equations for \left\langle1_{x}|1_{y}\right\rangle we already know that this state is orthogonal to each others because 1 state at x-axis while the others in y-axis for...
  26. R

    Dimension of subspace of V^n with orthogonal vectors

    in a space V^n, prove that the set of all vectors {v1,v2,..}, orthogonal to any v≠0, form a subspace V^(n-1). i know that a subspace of V^n must be at least one dimension less and the set of vector v1,v2,... build a orthogonal basis, but how can one show with this preconditions that the...
  27. T

    Finding an Orthogonal Vector and Calculating Triangle Area from Given Points

    Homework Statement Find a nonzero vector orthogonal to the plane through points P (0, -2, 0) Q (4, 1, -2) and R (5,3,1) and find the area of the triangle formed by PQR. The attempt at a solution To be honest, I am not entirely sure how to do this problem. I've looked through my textbook...
  28. T

    What is the so called orthogonal operator basis ?

    What is the so called "orthogonal operator basis"? What is the so called "orthogonal operator basis"?
  29. P

    Question about definite integral and orthogonal functions

    Hello, I am just going through a book on calculus and understand that the definite integral can be interpreted as area under the curve. Now I am trying to figure out the orthogonality relationship between functions and this is normally defined (as far as I can tell from the internet resources)...
  30. D

    Is There a Name for the Decomposition of a Partitioned Orthogonal Matrix?

    "Partitioned Orthogonal Matrix" Hi, I was reading the following theorem in the Matrix Computations book by Golub and Van Loan: If V_1 \in R^{n\times r} has orthonormal columns, then there exists V_2 \in R^{n\times (n-r)} such that, V = [V_1V_2] is orthogonal. Note that...
  31. S

    Definition of Orthogonal Matrix: Case 1 or 2?

    Is the definition of an orthogonal matrix: 1. a matrix where all rows are orthonormal AND all columns are orthonormal OR 2. a matrix where all rows are orthonormal OR all columns are orthonormal? On my textbook it said it is AND (case 1), but if that is true, there's a problem: Say...
  32. B

    Orthogonal projection of 2 points onto a plane

    edit: This thread might need moved, sorry about that. Hi, I have ended up on this site a few times after searching various maths issues; it seems to have a good community so I am asking you good people for a little help understanding this. Tomorrow I have a semi-important maths exam, if I fail...
  33. K

    Orthogonal projection onto line L

    [b] Def1. Let L be a line in E. We define the "orthogonal projection onto L" to be Ol = {(P,Q)| P,Q in E and either 1.P lies on L and P=Q or 2.Q is the foot of the perpendicular to L through P. Problem 1. Let L be a line in E. Show that Ol is not a rigid motion because it fails...
  34. P

    Prove l*conj(l)=1 for Orthogonal Matrix A

    Homework Statement Let l be an eigenvalue of an orthogonal matrix A, where l = r + is. Prove that l * conj(l) = r^2 + s^2 = 1. Homework Equations The Attempt at a Solution I am really confused on where to go with this one. I have Ax = A I x = A A^T A x = l^3 x and Ax = l...
  35. M

    Load sharing in parallel vs orthogonal screws

    Hi I would be grateful for some help or pointers for the following question. I am an orthopaedic surgeon and often when we fix fractures we use screws to hold the bone in place. We use different configurations of screws (ie one or two parallel or orthogonal, two screws at right angles to...
  36. Z

    A courious thing about orthogonal polynomials

    given a set of orthogonal polynomials p_{n} (x) with respect to a certain positive measure \mu (x) > 0 on a certain interval (a,b) then i have notices for several cases that f(z) defined by the integral transform \int_{a}^{b}dx\mu(x)cos(xz)=f(z) has ALWAYS only real roots ¡¡ *...
  37. Y

    Question on orthogonal function with respect to weight.

    If F(x) and G(x) is orthogonal with respect to weight W(x), does this mean F(x) and G(x) are not necessary orthogonal by themselves? \intF(x)G(x)W(x)dx=0 do not mean \intF(x)G(x)dx=0 If \intF(x)G(x)dx=0 then W(x)=1 Thanks Alan
  38. S

    Show that this orthogonal diagonalization is a singular value decomposition.

    Homework Statement Prove that if A is an nxn positive definite symmetric matrix, then an orthogonal diagonalization A = PDP' is a singular value decomposition. (where P' = transpose(P))2. The attempt at a solution. I really don't know how to start this problem off. I know that the singular...
  39. R

    Linear algebra proof - Orthogonal complements

    Homework Statement Let V be an inner product space, and let W be a finite dimensional subspace of V. If x is not an element of W, prove that there exists y in V such that y is in the orthogonal complement of W, but the inner product of x and y is not equal to 0. Homework Equations The...
  40. N

    Are Ho(x) and H1(x) orthogonal to H2(x) with respect to e^(-x^2)?

    In my third year math class we were asked a question to prove that Ho(X) and H1(x) are orthogonal to H2(x), with respect to the weight function e^(-x^2) over the interval negative to positive infinity where Ho(x) = 1 H1(x) = 2x H2(x) = (4x^2) - 2 i know that i have to multiply Ho(x) by...
  41. Rasalhague

    Orthogonal Transformations

    In Chapter 1 of Blandford & Thorne: Applications of Classical Physics, section 1.7.1, "Euclidean 3-space: Orthogonal Transformations" (Version 0801.1.K), do equations 1.43 at the beginning of the section, representing respectively the expansion of the old basis vectors in the new basis, and the...
  42. J

    Vector expressed in a basis noncoplanar, neither orthogonal nor of unit length

    We have three orthonormal vectors \vec i_1 , \vec i_2, \vec i_3 , and we know which are the components of an arbitrary vector \vec A in this base, explicitly: \vec A = (\vec A \bullet \vec i_1) \vec i_1 + (\vec A \bullet \vec i_2) \vec i_2 + (\vec A \bullet \vec i_3) \vec i_3...
  43. Z

    Do Orthogonal Polynomials have always real zeros ?

    Do Orthogonal Polynomials have always real zeros ?? the idea is , do orthogonal polynomials p_{n} (x) have always REAl zeros ? for example n=2 there is a second order polynomial with 2 real zeros if we consider that there is a self-adjoint operator L so L[p_{n} (x)]= \mu _{n} p_{n} (x)...
  44. W

    Finding Parallel and Orthogonal Vectors for u and v

    Homework Statement For u=(26, 6, 21) and v=(−27, −9, −18) , find the vectors u1 and u2 such that: (i) u1 is parallel to v (ii) u2 is orthogonal to v (iii) u = u1 + u2 Homework Equations None The Attempt at a Solution I'm quite lost on this question and not sure...
  45. B

    Solving Orthogonal Matrix Homework w/ Symmetric Matrix

    Homework Statement Given the symmetric Matrix 1 2 2 5 find an orthogonal matrix P such that C=BAB^t Homework Equations The Attempt at a Solution I found the eigenvalues to be 3-(2\sqrt{2}) and 3+(2\sqrt{2}) giving eigenvectors of [1,1-\sqrt{2}] and...
  46. B

    Normalizing an Orthogonal Basis: A Step-by-Step Guide

    Homework Statement I have used the gram schmidt process to find an orthogonal basis for {1,t,t^2} which is (1,x,x^2 - \frac{2}{3}) How to i normalize these Homework Equations e_1=\frac{u_1}{|u_1|} The Attempt at a Solution...
  47. K

    How do i find the orthogonal projection of a curve?

    Homework Statement curve S is the intersections of two surfaces, i have to find the curve obtained as the orthogonal projection of the curve S in the yz-planeHomework Equations how do i find the orthogonal projection of curve S??The Attempt at a Solution i found the equation of curve S to be...
  48. J

    Orthogonal matrix whose submatrix has special properties

    Dear Forumers. I am working on the following problem. Let matrix P=( A B ) where A and B are matrices. Let P be an n*n orthogonal matrix. Show that A'A is an idempotent matrix. I do not know where to start. Thanks in advance for the help.
  49. J

    Orthogonal vector spaces and matrices

    Hi everyone, I would need to get some help on the following question Let A (m*n) Let B (m*p) Let L(A) be the span of the columns of A. L(A) is orthogonal to L(B) <=> A'B=0 I suppose that the => direction is pretty obvious, since A is in L(A) and B in is L(B). Now I am not sure how to...
  50. M

    How Do Orthogonal Projectors Influence Dimensions and Norms?

    Let P and Q be two m x m orthogonal projectors. We show a) ||P-Q||_2 <or eq. 1 b)||P-Q||_2 < 1 implies the ranges of P and Q have equal dimensions. I think I must use the properties of orthogonal projectors. I guess Range(P) Inters Null(P) = {0} and Range(Q) Inters...
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