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. W

    Orthogonal matrix and eigenvalues

    a) Let M be a 3 by 3 orthogonal matrix and let det(M)=1. Show that M has 1 as an eigenvalue. Hint: prove that det(M-I)=0. I think I'm supposed to begin from the fact that det(M)=1=det(I)=det(MTM) and from there reach det(M-I)=0 which of course would mean that there's an eigenvalue of 1 as...
  2. M

    Finding a Basis for the Orthogonal Complement of W

    Homework Statement Let W be the subspace of R^3 spanned by the vectors v1=[2,1,-2] and v2=[4,0,1] Find a basis for the orthogonal complement of W Homework Equations None The Attempt at a Solution I can do this question except for the fact when i get the matrix in form...
  3. P

    Linearly independent but not orthogonal, how come?

    Hi everyone, I was reading about Gram-Schmidt process of converting two linearly independent vectors into orthogonal basis. But, as I understand, if two vectors are linearly independent then they are orthogonal! isn't that right?? Could anybody explain... please.
  4. Saladsamurai

    Show the product of two Orthogonal Matrices of same size is Orthogonal

    Homework Statement Show that the product of two Orthogonal Matrices of same size is an Orthogonal matrix. I am a little lost as to how to start this one. If A is some nxn orthogonal matrix, than AA^T=A^TA=I where I is the nxn identity matrix. So now what? Let's take A and B to be nxn...
  5. nicksauce

    Understanding Orthogonal Integral on Introduction to Quantum Mechanics

    On page 102 of Introduction to Quantum Mechanics, Griffiths writes that \int_{-\infty}^{\infty}e^{i\lambda x}e^{-i \mu x}dx = 2\pi\delta(\lambda-\mu) I don't see how this follows. If you replace lambda with mu, then you get a cancellation, and the integral of 1dx. Am I missing something?
  6. rocomath

    Orthogonal Eigenvector, Proof is bothering me

    Suppose A\overrightarrow{x}=\lambda_1\overrightarrow{x} A\overrightarrow{y}=\lambda_2\overrightarrow{y} A=A^T Take dot products of the first equation with \overrightarrow{y} and second with \overrightarrow{x} ME 1) (A\overrightarrow{x})\cdot...
  7. E

    Orthogonal Matrix: Column or Row Vectors?

    Homework Statement My book used the term "orthogonal 3 by 3 matrix" and I couldn't find where that was defined. Does that mean that the column vectors are orthogonal or that the row vectors are orthogonal? Or are those two things equivalent? Homework Equations The Attempt at a Solution
  8. M

    Quick Question: Is this matrix an orthogonal projection?

    [SOLVED] Quick Question: Is this matrix an orthogonal projection? Homework Statement P=[0 0 ] [11] Homework Equations The Attempt at a Solution Its orthogonal if the null space and range are perpendicular. Range=[0 ] [x+y] null space=[x
  9. S

    Linear Algebra - Orthogonal Projections

    Homework Statement Prove that if P in L(V) is such that P2 = P and every vector in Ker(P) is orthogonal to every vector in Im(P), then P is an orthogonal Projection. Homework Equations Orthogonal projections have the following properties: 1) Im(P) = U 2) Ker(P) = Uperp 3) v - P(v)...
  10. E

    Is the trace of a matrix preserved by an orthogonal transformation?

    Homework Statement My statistical mechanics book says that if M is an real, symmetric n by n matrix, and U is the matrix of its eigenvectors as column vectors, then the transformation UMU^{-1} preserves the trace of M. Is that true? If so, is it obvious? If it is true but not obvious, how do...
  11. A

    Orthogonal Diagonalization of a Symmetric Matrix

    Homework Statement Orthogonally diagonalize the matrix: | 2 1 1| | 1 2 1| | 1 1 2| Homework Equations Since this only has...
  12. tony873004

    Website title: Are These Vectors Orthogonal, Parallel, or Neither?

    Homework Statement Determine whether the given vectors are orthogonal, parallel, or neither. Homework Equations \cos \theta = \frac{{\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} }}{{\left| {\overrightarrow {\rm{a}} } \right|\left| {\overrightarrow {\rm{b}} } \right|}}\,\...
  13. quasar987

    Hilbert space & orthogonal projection

    [SOLVED] Hilbert space & orthogonal projection Homework Statement Let H be a real Hilbert space, C a closed convex non void subset of H, and a: H x H-->R be a continuous coercive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M \geq 0 such that |a(x,y)| \leq...
  14. S

    What are the possible values of the determinant of an orthogonal matrix?

    Hi I had a final today and one of the questions was find all the possible values of det Q if Q is a orthogonal matrix I m still wondering how would I do this? Any ideas?
  15. M

    How to prove the orthogonality of eigenfunctions in Sturm-Louisville problems?

    [SOLVED] orthogonal eigenfunctions From Sturm-Louisville eigenvalue theory we know that eigenfunctions corresponding to different eigenvalues are orthogonal. For example, \Phi_{xx} + \lambda \Phi = 0 would be of Sturm-Louisville form (note: \Phi_{xx} represents the second derivative of...
  16. B

    Ind the orthogonal trajectories of the family of curves

    Homework Statement Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y = x/(1+kx) 2. The attempt at a solution I have been trying this problem for hours, and I get a different answer every time...
  17. A

    Finding the Value of C for Orthogonal Intersection of Two Curves

    Homework Statement Two curves intersect orthogonally when their tangent lines at each point of intersection are perpendicular. Suppose C is a positive number. The curves y=Cx^2 and y=(1/x^2) intersect twice. Find C so that the curves intersect orthogonally. For that value of C, sketch both...
  18. E

    Showing two families of curves are orthogonal.

    Let the function f(z) = u(x,y) + iv(x,y) be analytic in D, and consider the families of level curves u(x.y)=c1 and v(x,y)=c2 where c1 and c2 are arbitrary constants. Prove that these families are orthogonal. More precisely, show that if zo=(xo,yo) (o is a subscript) is a point in D which is...
  19. T

    Understanding Orthogonal & Linear Combinations

    I was reading the literature, and come across this. But I couldn't decipher what it means by linear combination and orthogonal combination. It is sufficient to consider two axionic fields, A and B, with a potential: V = \lambda_{1}^4 [1 - cos(\frac{\theta}{f_1} + \frac{\rho}{g_1})] +...
  20. C

    Orthogonal Matrices Explained: Examples & More

    what exactly are orthogonal matrices? can someone give me an example of how they would look like?
  21. 4

    What is the Orthogonal Decomposition of x from b in terms of RowA and NulA?

    Let A= [ 1 -2 -1 2] [-1 0 3 -2 ] [ 3 -4 -5 6] (sorry, can't line up the columns) I think I've done a) and b) correctly, I don't really understand c), d) and e) a) Find RowA and Nul A RowA={( 1, -2, -1, 2), (0, 1, -1, 0)} Nul A= {(3, 1, 1, 0), (-2, 0, 0, 1)} b) If...
  22. daniel_i_l

    Orthogonal transformations

    Homework Statement I have a general question. If we have some subspace W of R^n where dimW=k. Then if T is an orthogonal transformation from R^n->R^n is the dimension of T(W) also k? Homework Equations The Attempt at a Solution The reason I think this is true is because if...
  23. J

    Regarding Orthogonal Transformations

    Find an orthogonal transformation T from R3 to R3 such that T of the column vector [2/3 2/3 1/3] is equal to the column vector [0 0 1] So I tried to construct out the 3x3 matrix [a b c] [d e f] [g h i] and applied the properties of an orthogonal matrix and basic algebra. I ended up with a...
  24. L

    The significance of orthogonal relationships

    what is the meaning of orthogonal relationships in addition to right angles in the xyz coordinate system? for instance, if a 3 piece rocket separated in space in an orthogonal way...will there be any significance when compared to the 3 piece rocket that does not separate in an orthogonal...
  25. L

    Finding the Orthogonal Projection of a Vector onto Another Vector

    Given: \vec A \cdot \vec B = non zero and \theta does not equal 0 I can't seem to prove that Vector B minus the Projection of B onto A makes the orthogonal projection of B onto A. Can you help?
  26. daniel_i_l

    Existence of Orthogonal Transformation for Given Sub-spaces in R^n

    Homework Statement Given two sub-spaces of R^n - W_1 and W_2 where dimW_1 = dimW_2 =/= 0. Prove that there exists an orthogonal transformation T:R^n -> R^n so that T(W_1) = T(W_2) Homework Equations The Attempt at a Solution If dimW_1 = dimW_2 = m then we can say that...
  27. N

    Orthogonal Matrices: Rotation & Reflection

    We know that if M is an orthogonal matrix,then DetM=(+-)1 When Det M=1,thee transformation is a rotation.And for reflection about anyone o all three axes DetM=-1. I did this.. But I did not know that information:When Det M=1,thee transformation is a rotation.And for reflection about anyone...
  28. Y

    Find the orthogonal projection

    Homework Statement My questions is this: How to find the orthogonal projection of vector y= (7,-4,-1,2) on null space N(A) Where A is a matrix A = \left(\begin{array}{cccc}2&1&1&3\\3&2&2&1\\1&2&2&-9\end{array}\right) Homework Equations A^TA\overline{x}=A^T\overline{y} The...
  29. R

    Real Eigenvalues and 3 Orthogonal Eigenvectors for Matrix (c,d)

    Homework Statement For which real numbers c and d does the matrix have real eigenvalues and three orthogonal eigenvectors? 120 2dc 053 Homework Equations im having trouble getting started on this one. Ive tried using solving for the eigenvalues pretending that c and d are...
  30. P

    Electric and magnetic waves orthogonal to each other?

    In my intro to E&M course, in the section on electromagnetic waves, my textbook just says that electric and magnetic waves are orthogonal to each other, but it doesn't say why. How do we know this? Is it from solving the wave partial differential equation? If so, given that I've tooken a course...
  31. V

    Linear Algebra - Orthogonal Vectors

    I'm a bit confused, conceptually. This is the problem Let v1=( 1, -1, 2) v2=( 2, 1, 3) v3=( 1, -4, 3) Find a nonzero vector u that is orthogonal to all three vectors v1, v2, and v3. I know how to find the projection matrix, P, which I can solve with v1, v2, and v3. The equation for that is...
  32. T

    Symmetric matrices and orthogonal projections

    Homework Statement Consider a symmetric n x n matrix A with A² = A. Is the linear transformation T(x) = Ax necessarily the orthogonal projection onto a subspace IR^n? Homework Equations The Attempt at a Solution No idea what thought to begin with.
  33. M

    Does an Improper Orthogonal Matrix Always Have a Determinant of -1?

    The question is (true or false) if Q is an improper 3 x 3 orthogonal matrix then Q^2 = I. The way I have approached it so far has been a brute force method. I'm not really sure if this will be true or false, and I have a feeling it is false, but I can't construct a good counter-example...
  34. M

    Orthogonal unit vectors also unit vectors?

    If two vectors v, w are both unit vectors, then v+w and v-w will be orthogonal, but are v+w and v-w also unit vectors? I would say no because the inner product of the two added, and the two subtracted would also have to be orthogonal. <(v+w)+(v-w),(v+w)-(v-w)> = <2v,2w> and <2v,2w> must be...
  35. N

    Eigenfunctions of Excited States: Why Orthogonal?

    Why is it that eigenfunctions of different excited states for 1 atom have to be orthogonal?
  36. D

    Family of orthogonal trajectories for a vertical parabola

    1. Homework Statement . The correct answer is E 2. Homework Equations :Procedure from our text: "Step 1. Determine the differential equation for the given family F(x, y,C) = 0. Step 2. Replace y' in that equation by −1/y'; the resulting equation is the differential equation for the family of...
  37. S

    What is the Equation for Finding Orthogonal Trajectories?

    Homework Statement Write the equation to find the orthogonal trojectory for x^2 + y^2 = r^2 2. The attempt at a solution Well.. you solve this by just integrating.. but, I don't know what to do with the r^2 ? x^2 + y^2 = r^2 x^2 + y^2 \frac{dy}{dx} = r^2 \frac{dy}{dx} is that...
  38. V

    Forming an orthogonal matrix whose 1st column is a given unit vector

    Homework Statement Show that if the vector \textbf{v}_1 is a unit vector (presumably in \Re^n) then we can find an orthogonal matrix \textit{A} that has as its first column the vector \textbf{v}_1. The Attempt at a Solution This seems to be trivially easy. Suppose we have a basis \beta for...
  39. S

    Proving Orthogonal Matrix with Identity Matrix and Non-Zero Column Vector a

    Assume that I is the 3\times 3 identity matrix and a is a non-zero column vector with 3 components. Show that: I - \frac{2}{| a |^{2}}aa^{T} is an orthogonal matrix?My question is how can one take the determinant of a if it is not a square matrix? Is there a flaw in this problem? Thanks
  40. M

    Proving Orthogonal Curves: y1=-.5x^2+k & y2=lnx+c

    We were given a graded assigment and one of the question asks. Prove that all curves in the family y1=-.5x^2 + k (k any constant) are perpendicular to all curves in the family y2=lnx+ c (c any constant) at their points of intersection. I found the derivatives of y1 and y2 and they are...
  41. V

    Finding Orthogonal Projection of \overrightarrow{x}

    Question (5.1, #26 -> Bretscher, O.): Find the orthogonal projection of \left[\begin{array}{c} 49 \\ 49 \\ 49 \end{array}\right] onto the subspace of \mathbb{R}^3 spanned by \left[\begin{array}{c} 2 \\ 3 \\ 6 \end{array}\right] and \left[\begin{array}{c} 3 \\ -6 \\ 2 \end{array}\right]...
  42. L

    Orthogonal Functions: Questioning the Reason Why

    A question for my understanding: If I have an operator \cal{L} and a set of eigenfunctions \phi_n of this operator, then the eigenfunctions are orthogonal. Why is that?
  43. S

    Proving Orthogonal Functions: Integral of $\phi_{m}^* \phi_{n}$

    In the first hald of this question it was proven that -\frac{\hbar^2}{2m} \frac{d}{dx} \left[ \phi_{m}^* \frac{d \phi_{n}}{dx} - \phi_{n} \frac{d \phi_{m}^*}{dx}\right] = (E_{m} - E_{n}) \phi_{m}^* \phi_{n} By integrating over x and by assuming taht Phi n and Phi m are zero are x = +/-...
  44. P

    Orthogonal Trajectory

    I have to find the orthogonal trajectory to the family of circles tangents to the x-axis. I eventually found that the derivative of the orthogonal trajectory would be: \frac{dy}{dx}= \frac{-x^2 + y^2}{2xy} But how do i find the equation for "y" explicitly? (ps. This is not an assignment...
  45. N

    Inner products and orthogonal basis

    Hi all! This looks a pretty nice forum. So here's my question: How do I find/show a basis or orthogonal basis relative to an inner product? The reason I ask, is because in my mind I see the inner product as a scalar, and thus I find it difficult to "imagine" how a scalar lives in a...
  46. S

    General Orthogonal Coordinate System: Line Element Explained

    A general orthogonal coordinate system (u,v,w) will have a line elemet of the form: ds^2 = f^2 du^2 + g^2dv^2 + h^2dw^2 I have done a lot of vector calculus, but for some reason I can't figure out what this means! What is a line element? I know about the differential length element and its...
  47. F

    Dot product of basis vectors in orthogonal coordinate systems

    I'm doing a series of questions right now that is basically dealing with the dot and cross products of the basis vectors for cartesian, cylindrical, and spherical coordinate systems. I am stuck on \hat R \cdot \hat r right now. I'll try to explain my work, and the problem I am running into...
  48. M

    Finding the Orthogonal Trajectory of x^p + Cy^p = 1

    I am working on this problem, and have a simple question. Determine the orthogonal trajectory of x^p + Cy^p = 1 where p = constant. I start out by taking the derivative with respect to x. My question is this. does Cy^p become Cpy^{p-1} or C_1y^{p-1} ? Thanks, Morgan
  49. O

    How do I transform a cross product using an orthogonal matrix?

    I've been banging my head against this problem for some time now, and I just can't solve it. The problem seems fairly simple, but for some reason I don't get it. Given the coordinate transformation matrix A=\left(...
  50. H

    What is the Matrix for Orthogonal Projection to the xy-plane?

    After 10 years of teaching middle school, I am going back to grad school in math. I haven't seen Linear Algebra in more than a decade, but my first class is on Generalized Inverses of Matrices (what am I thinking?). I have a general "rememberance" understanding of most of the concepts we're...
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