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. ognik

    MHB Did I Make a Mistake in Finding the Determinant of an Orthogonal 2x2 Matrix?

    Find the general form of an orthogonal 2 x 2 matrix = $ \begin{bmatrix}a&b\\c&d\end{bmatrix}$ I used $det(A)=\pm 1$ (from an earlier exercise) - and the special form of $ A_{2 \times 2}^{-1} = \frac{ \begin{bmatrix}d&-b\\-c&a\end{bmatrix}}{|A|} $ using $|A| = +1$ first, to get an...
  2. S

    Orthogonal matrices form a group

    Homework Statement Show that the set of all ##n \times n## orthogonal matrices forms a group. Homework Equations The Attempt at a Solution For two orthogonal matrices ##O_{1}## and ##O_{2}##, ##x'^{2} = x'^{T}x' = (O_{1}O_{2}x)^{T}(O_{1}O_{2}x) = x^{T}O_{2}^{T}O_{1}^{T}O_{1}O_{2}x =...
  3. B

    Describe all vectors orthogonal to col(A) with a twist

    I am trying to solve the following problem: Let A be a real mxn matrix. Describe the set of all vectors in F^m orthogonal to Col(A). Here, F^m could be C^m. Now in the real case, I'd say that the column space of A is the row space of A^T, and it is well known that the row space of a matrix is...
  4. S

    Invariance of quadratic form for orthogonal matrices

    Homework Statement Show that all ##n \times n## (real) orthogonal matrices ##O## leave invariant the quadratic form ##x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}##, that is, that if ##x'=Ox##, then ##x'^{2}=x^{2}##. Homework Equations The Attempt at a Solution ##x'^{2} = (x')^{T}(x') =...
  5. S

    Are the E and H field not orthogonal near the antenna?

    Hello, I've heard in many places that in the "near reactive field" of an antenna which is the region really really close to the antenna, the E and H fields are not perpendicular. But I just can't imagine how that is possible since In Maxwell's 3'rd and 4'th equations it is explicit that the curl...
  6. M

    Given a vector, how to compute orthogonal plane

    Given a vector (in 3-d), how do I determine the plane that is orthogonal to it? I am not quite finding a search term that gets me to this, but instead to several similar, but different questions. One such is find an equation of a plane perpendicular to a vector and passing through a given...
  7. brotherbobby

    Proving "Rotation Matrix is Orthogonal: Necessary & Sufficient

    I'd like to prove the fact that - since a rotation of axes is a length-preserving transformation, the rotation matrix must be orthogonal. By the way, the converse of the statement is true also. Meaning, if a transformation is orthogonal, it must be length preserving, and I have been able to...
  8. T

    How to construct a vector orthogonal to all but one?

    Given n linearly independent vectors, v1, v2, v3, ...vn. How to find construct a vector that is orthogonal to v2, v3, ..., vn (all v but not v1)? Is Gram Schmitt process the way to do this? or just by brute force?
  9. N

    Orthogonal properties of confluent hypergeometric functions

    Hi Can anyone point to me a reference where orthogonal properties of confluent hypergeometric functions are discussed? Navaneeth
  10. SSGD

    Convert Perspective picture to Ortho picture for CAD

    I want to take a picture of an object. Then import the image into cad. And trace it. The issue I am having is the picture from the camera is a 3D perspective image. So it is impossible for me to trace in CAD. Is there a way to convert an image from perspective to orthogonal? Or create a...
  11. nuuskur

    Find all orthogonal matrices in R

    Homework Statement Assuming I understand the problem correctly, I need to define the set of all orthogonal matrices. Homework Equations The Attempt at a Solution Per the definition of orthogonal matrix: Matrix ##A\in Mat_n(\mathbb{R})## is orthogonal if ##A^tA = I## If ##O## is the set of all...
  12. RJLiberator

    Orthogonal and Diagonal Matrices

    Homework Statement Find all 2 x 2 and 3 x 3 orthogonal matrices which are diagonal. Construct an example of a 3 x 3 orthogonal matrix which is not diagonal. Homework Equations Diagonal Matrix = All components are 0 except for the diagonal, for a 2x2 matrix, this would mean components a and d...
  13. M

    Dot Products With Orthogonal Vectors

    Homework Statement The Vector a= -2i -3j and is orthogonal to vector b that has the same length as a. The third vector c has the dot products ca= 8m^2 and cb= 9m^2. What are the components of c? c = ______i + _____ j m Homework EquationsThe Attempt at a Solution I know that (a⃗ +b⃗ )⋅c⃗ =a⃗...
  14. Engineerbrah

    Whats the point of a function being orthogonal?

    I understand that a function is orthogonal if the inner product of any two functions of an infinite series equal to zero. My question is why do we prove functions are orthogonal? What can we do with this information?
  15. T

    MHB Tschirnhausen Curve: Finding Tangents at a Given Point

    Are the given families orthogonal trajectories of each other? {x}^{2}+{y}^{2} = ax {x}^{2}+{y}^{2} = by I first started by finding them implicitly. \frac{2x+a}{2y} = y' \frac{2x+b}{2y} = y' Then the problem wanted me to sketch my answer. The Tschirnhausen, I solved. I just would...
  16. B

    Finding Coefficients of Orthogonal Quadric Equations

    Given a quadric equation (F(x,y) = 0), exist other quadric equation (G(x,y) = 0) such that the poinst of intersection between the graphics are ortogonals. So, how to find the coefficients of the new quadic equation? EDIT: I think that F and G needs to satisfy∇F • ∇G = 0. So, if F is known, how...
  17. JesseJC

    Orthogonal Projection and Reflection: Finding the Image of a Point x = (4,3)

    Homework Statement |-1/2 -sqrt(3)/2 | |sqrt(3)/2 -1/2 | Homework Equations I don't know The Attempt at a Solution Hey everyone, I've been asked to find the "orthogonal projection" on this matrix, this is part B to a...
  18. ELB27

    Proving a certain orthogonal matrix is a rotation matrix

    Homework Statement Let ##U## be a ##2\times 2## orthogonal matrix with ##\det U = 1##. Prove that ##U## is a rotation matrix. Homework EquationsThe Attempt at a Solution Well, my strategy was to simply write the matrix as $$U = \begin{pmatrix} a & b\\ c & d \end{pmatrix}$$ and using the given...
  19. D

    Sum of Two Periodic Orthogonal Functions

    Homework Statement This problem is not from a textbook, it is something I have been thinking about after watching some lectures on Fourier series, the Fourier transform, and the Laplace transform. Suppose you have a real valued periodic function f with fundamental period R and a real valued...
  20. B

    Is matrix hermitian and its eigenvectors orthogonal?

    I calculate 1) ##\Omega= \begin{bmatrix} 1 & 3 &1 \\ 0 & 2 &0 \\ 0& 1 & 4 \end{bmatrix}## as not Hermtian since ##\Omega\ne\Omega^{\dagger}## where##\Omega^{\dagger}=(\Omega^T)^*## 2) ##\Omega\Omega^{T}\ne I## implies eigenvectors are not orthogonal. Is this correct?
  21. evinda

    MHB Finding Complete Orthogonal Systems for Boundary Value Problems

    Hello! (Wave) Suppose that we have a $C^{\infty}$ function $f: [0, \pi] \to \mathbb{R}$ for which it holds that $f(0)=f(\pi)=0$. How can we find a complete orthogonal system of this space? (Thinking)
  22. S

    Double Orthogonal Closed Subspace Inner Product => Hilbert

    Let X be an Inner Product Space. If for every closed subspace M, M^{\perp \perp} = M, then X is a Hilbert Space (It's complete). Hint: Use the following map: T : X \longrightarrow \overset{\sim}{X}: T(y)=(x,y)=f(x) where (x,y) is the inner product of X. Relevant equations: S^{\perp} is always...
  23. SU403RUNFAST

    Differential equations, orthogonal trajectories

    Homework Statement you are given a family of curves, in this case i was given a bunch of circles x^2+y^2=cx, sketch these curves for c=0,2,4,6, both positive and negative, solve the equation for c and differentiate both sides with respect to x and solve for dy/dx. You obtain an ODE in the form...
  24. B

    Gradient Vector is Orthogonal to the Level Curve

    Homework Statement Let f(x,y)=arctan(x/y) and u={(√2)/2,(√2)/2} d.) Verify that ∇fp is orthogonal to the level curve through P for P=(x,y)≠(0,0) where y=mx for m≠0 are level curves for f. Homework Equations The Attempt at a Solution ∇f={(y)/(x^2+y^2),(-y)/(x^2+y^2)} m=1/tan(k) where...
  25. F

    Orthogonal complement of the intersection of 2 planes

    Homework Statement Let W be the intersection of the two planes: x-y+z=0 and x+y+z=0 Find a basis for and the dimension of the orthogonal complement, W⊥ Homework EquationsThe Attempt at a Solution The line x+z=0 intersects the plane, which is parameterized as t(1, 0, -1) Then W⊥ is the plane...
  26. R

    What the terms orthogonal & basis function denote in case of signals

    I am a beginer. I have read that any given signal whether it simple or complex one,can be represented as summation of orthogonal basis functions.Here, what the terms orthogonal and basis functions denote in case of signals? Can anyone explain concept with an example?Also,what are the physical...
  27. onethatyawns

    How many orthogonal spins can an object have?

    I'm not talking about particles with defined spins. Imagine something visible to the human eye. It can spin in one direction (obviously), it can spin in a perpendicular dimension (momentarily before these two spins begin to combine), and I'm wondering if more spins can be added on this system...
  28. ognik

    MHB Antisymmetry Invariant Under Similarity Orthogonal Transforms

    Hi - the text is very brief on similarity transforms and wiki etc. a bit beyond where I am. In fact I think I am muddling a few things up, so I have a few questions around this topic please: 1) I'd appreciate a 'beginners' explanation of similarity transforms, what they really are and what they...
  29. ognik

    Orthogonal coordinate systems - scale factors

    Homework Statement Start from the 'relevant equation' below and derive $$ (1) \frac{\partial{\bf{\hat{q}}_{i}}}{{\partial{q}}_{j}}={\hat{q}}_{i}\frac{1}{{h}_{i}} \frac{\partial{h}_{i}}{{\partial{q}}_{j}}, {i}\ne{j}$$ $$ (2) \frac{\partial{\bf{\hat{q}}_{i}}}{{\partial{q}}_{i}}= -\sum...
  30. T

    Velocity Addition for Orthogonal Boosts in SR

    Hi, is there any general formula to find out the final velocity w, happened by a boost in x direction forst and then to y direction? I could find the boost matrices for both and I know it's not purely another boost, rather a boost and a rotation, but I am really confused which formula to use...
  31. ognik

    MHB How Do You Prove (det A)aij = Cij(A) for an Orthogonal Matrix with det(A)=+1?

    Hi, the question (from math methods for physicists) is: If A is orthogonal and det(A)=+1, show that (det A)aij = Cij(A). I know that if det(A)=+1, then we are looking at a rotation. (Side question - I have seen that det(A) =-1 can be a reflection, but is 'mostly not reflections'; what does...
  32. ognik

    MHB General orthogonal scale factor identity

    Please be patient as I struggle with latex here ... Part 1 of the problem says to start with: $ \frac{\partial\bar{r}}{\partial{q}_{1}} ={h}_{1} \hat{q}_{1} $ and then to find an expression for $ {h}_{1} $ that agrees with $ {g}_{ij}=\sum_{l}...
  33. P

    Find all orthogonal 3x3 matrices of the form

    Homework Statement Find all orthogonal 3x3 matrices of the form \begin{array}{cc} a & b & 0 \\ c & d & 1\\ e & f & 0 \\\end{array} Homework Equations There are many properties of an orthogonal matrix. The one I chose to use is: An n x n matrix is an orthogonal matrix IFF $$A^{T}A = I$$. That...
  34. gfd43tg

    Eigenfunctions orthogonal in Hilbert space

    Hello, I am having a question regarding how eigenfunctions are orthogonal in Hilbert space, or what does that even mean (other than the inner product is zero). I mean, I know in ##\mathbb {R^{3}}##, vectors are orthogonal when they are right angles to each other. However, how can functions be...
  35. Math Amateur

    MHB Issue 2 - Tapp - Characterizations of the Orthogonal Groups

    I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 2 in Chapter 3, namely: "2. Several Characterizations of the Orthogonal Groups". I need help in fully understanding the proof of Proposition 3.10. Section 2 in Ch. 3...
  36. Math Amateur

    MHB Characterizations of the Orthogonal Groups _ Tapp, Ch. 3, Section 2

    I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focussed on and studying Section 2 in Chapter 3, namely: "2. Several Characterizations of the Orthogonal Groups". I need help in fully understanding some important remarks following Proposition 3.10...
  37. B

    Perfoming ANOVA test using orthogonal contrasts

    I have to test treatments B,C,D and E where ##H_0:\mu_B=\mu_C=\mu_D=\mu_E## vs ## H_1: not H_0##. Use ##\alpha=0.05## given this problem. Now ##\mu_A## is not included in the hypothesis so I am trying to figure out how to go about this problem. I was thinking of using orthogonal contrasts and...
  38. C

    Orthogonal projection and reflection (matrices)

    Homework Statement [Imgur](http://i.imgur.com/VFT1haQ.png) Homework Equations reflection matrix = 2*projection matrix - Identity matrix The Attempt at a Solution Using the above equation, I get that B is the projection matrix and E is the reflection matrix. Can someone please verify if this...
  39. P

    First order ODE, orthogonal trajectories

    1. The problem statement, all variables and given/known da ##\frac{x^{2}}{k^{2}} + \frac{y^{2}}{\frac{k^{4}}{4}} = 1## with k != 0 this can be simplified to ##x^{2} + 4y^{2} = k^{2}## Find dy/dx implicitly, then find the new dy/dx if you want orthogonal trajectories to the ellipse. Lastly solve...
  40. L

    How to show u x v in R^n space is orthogonal to u and v?

    I had a problem where I showed that u \times v in R^3 was orthogonal to u and v. I was wondering how I could show it for an R^n space? Like, what is the notation/expression to represent a cross product in an R^n space and how would I show that n-number of coordinates cancel out? Thank-you
  41. M

    Why is AB an Orthogonal Projection Matrix?

    I've attached the question to this post. The answer is false but why is it not considered the orthogonal projection? ## A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} ## ## B = \begin{bmatrix} x \\ y \end{bmatrix} ## ## AB =...
  42. J

    Duality and Orthogonality: What's the Difference?

    1. I cannot understand the difference between orthogonality and duality? Of course orthogonal vectors have dot product zero but isn't this the condition of duality as well? Swinging my head around it my cannot find the answer on the internet as well. 2.Relating to same concept is orthogonality...
  43. C

    Showing something satisfies Inner Product - Involves Orthogonal Matrices

    Homework Statement [/B] Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal. Show that the product <x, y> A = x · Ay is an inner product for R3. Homework Equations None The Attempt at a Solution I've shown that x · Dy is...
  44. _N3WTON_

    Orthogonal and Parallel Vectors

    Homework Statement For the following vectors 'u' and 'v', express 'u' as the sum u = p + n where 'p' is parallel to 'v' and 'n' is orthogonal to 'v' u = {-1, 2, 3} v = {2, 1, 1} Homework Equations Dot product Cross product The Attempt at a Solution First, I should say that I do not know how...
  45. D

    Sum of a vector parallel and orthogonal to.

    Homework Statement v = 3i - j u = 2i + j - 3k Express vector u as a sum of a vector parallel to v and a vector orthogonal to v.Homework Equations Proj of u onto v = [ (u • v) / |v|^2 ]v Expressing vector u as sum of a vector parappel to v and a vector vector orthogonal to v >> u = [Proj of...
  46. kq6up

    Solving a Diff. Eq. with Orthogonal Vectors

    I am working on a diff eq. that the prof. did as an example in class. ##y^{\prime\prime}-3y^{\prime}+2y=x^2+x+3## after subbing in I get: ##2a_2-6a_2x-3a_1+2a_2x^2+2a_1x+2a_o=x^2+x+3## She set aside the ##x^2## terms and set them equal to zero like such: ##2a_2x^2-x^2=0## I imagine this...
  47. C

    Lorentz transformation independence of axis orthogonal to velocity

    Apriori -- before taking any of the postulates of special relativity into account -- we might say that the lorentz transformations between two frames K and K', where K' is moving w. speed v along the x-axis of K, is given by $$\vec{x}' = F(\vec x, t)$$ and $$t' = G(\vec x, t).$$ Now, i want...
  48. I

    Linear Algebra orthogonal basis and orthogonal projection

    I was placed into honors calculus III for school. I was happy about this and I consider myself to be a pretty quick learner in math. However, my teacher is using many notations and terms that I am completely unfamiliar with. Mostly, I believe, because I've never taken linear algebra. I am...
  49. Math Amateur

    MHB Quick Question on Modules and Orthogonal Idempotents

    I am reading Berrick and Keating's book on Rings and Modules. Section 2.1.9 on Idempotents reads as follows: https://www.physicsforums.com/attachments/3097 https://www.physicsforums.com/attachments/3098So, on page 43 we read (see above) ... " ... ... Note that, conversely, a full set of...
  50. M

    Curve tangent is orthogonal to curve at a point

    Homework Statement . Let ##C## be a curve that doesn't pass through the origin and let ##P## be the closest point on the curve to the origin. Prove that the tangent to ##C## at ##P## is orthogonal to the vector ##P##. The attempt at a solution. Suppose ##P=\gamma(t_0)##, I want to...
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