Orthogonal Definition and 560 Threads

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

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\\...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

What is the so called "orthogonal operator basis"? What is the so called "orthogonal operator basis"?

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)...

"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...

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...

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...

[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...

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...

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...

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? \int[SIZE="5"]F(x)G(x)W(x)dx=0 do not mean \int[SIZE="5"]F(x)G(x)dx=0 If \int[SIZE="5"]F(x)G(x)dx=0 then W(x)=1 Thanks Alan

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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...

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.

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...

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...

Homework Statement Find a unit vector that is orthogonal to both i + j and i + k. I know I can solve this using the cross product of the two. But This chapter is about dot product and not cross product. I am not sure how I could go about solving this problem using the properties of...

Homework Statement i want to get the orthogonal trajectory of the curves of this family x^2 + y^2=cx Homework Equations answer is given as : y^2 + x^2=cy The Attempt at a Solution 2x + 2yy' = \frac {x^2 +y^2} {x} then y' = \frac{y} {2x} - \frac{x}{y} let v=y/x ...

How is it the determinant of an orthogonal matrix is \pm1. Is it: Suppose Q is an orthogonal matrix \Rightarrow 1 = det(I) = det(QTQ) = det(QT)det(Q) = ((det(Q))2 and if so, what is it for -1. Thanks.

The set of orthogonal trajectories for the family indicated by ( x-c)^2 + y^2 = c^2 My work: y' = -(x-c)/y Since c= ( x^2 + y^2 ) / 2x plugging back in and doing -1/y' i got y' = 2xy / ( x^2 - y^2) Then I am supposed to move the x and y to a side and integrate but i don't...

Homework Statement Let P be a projection. The definition used is P is a projection if P = PP. Show that ||P|| >=1 with equality if and only if P is orthogonal. Let ||.|| be the 2-normHomework Equations P = PP. P is orthogonal if and only if P =P*The Attempt at a Solution I've proved the...

Homework Statement Which of the following is the set of orthogonal trajectories for the family indicated by (x-c)^2 + y^2 = c^2 a). (x-c)^2 + y^2 = c^2 b). (x-c)^2 - y^2 = c^2 c). x^2 + (y-c)^2 = c^2 d). x^2 - (y-c)^2 = c^2 e). None of the above Homework Equations...

find u X v and show that it is orthogonal to both u and v. u= 6k v=-i + 3j + k http://s763.photobucket.com/albums/xx275/trinhkieu888/?action=view&current=666.jpg This is what I got from the picture, but my teacher said that I have one more step to do to show that they are orthogonal, I...

Homework Statement Given a state \mid \psi \rangle=\frac{1}{\sqrt{3}}[(i+1)\mid 1 \rangle + \mid 2 \rangle], find the normalized state \mid \psi^{'} \rangle orthogonal to to it.Homework Equations \langle \psi^{'} \mid \psi \rangle = 0 \langle \psi^{'} \mid \psi^{'} \rangle = 1The Attempt at...

Homework Statement S1 is in subspace of C^n. P unique orthogonal projector P : C^n -> S1, and x is in range of C^n. Show that the minimization problem: y in range of S1 so that: 2norm(x-y) = min 2norm(x-z) where z in range of S1 and variational problem: y in range of S1 so that...

Homework Statement P is mxm complex matrix, nonzero, and a projector (P^2=P). Show 2-norm ||P|| >= 1 with equality if and only if P is an orthogonal projector (P=P*) Homework Equations Let ||.|| be the 2-norm The Attempt at a Solution a. show ||P|| >= 1 let v be in the range...

I am somewhat confused about this property of an eigenvalue when A is a symmetric matrix, I will state it exactly as it was presented to me. "Properties of the eigenvalue when A is symmetric. If an eigenvalue \lambda has multiplicity k, there will be k (repeated k times), orthogonal...

Homework Statement Let A be an mxn matrix. a. Prove that the set W of row vectors x in R^m such that xA=0 is a subspace of R^m. b. Prove that the subspace W in part a. and the column space of A are orthogonal compliments. Homework Equations The Attempt at a Solution a. to...

Homework Statement Suppose P ∈ L(V) is such that P2 = P. Prove that P is an orthogonal projection if and only if P is self-adjoint.Homework Equations The Attempt at a Solution Let v be a vector in V. Let w be a vector in W and u be a vector in U and let U and W be subspaces of V where dim W+dim...

Suppose X\in\mathbb{R}^{n\times n} is orthogonal. How do you perform the computation of series \log(X) = (X-1) - \frac{1}{2}(X-1)^2 + \frac{1}{3}(X-1)^3 - \cdots Elements of individual terms are ((X-1)^n)_{ij} = (-1)^n\delta_{ij} \;+\; n(-1)^{n-1}X_{ij} \;+\; \sum_{k=2}^{n} (-1)^{n-k}...

Homework Statement Given the vectors u = (2, 0, 1, -4) v = (2, 3, 0, 1) Find any unit vector orthogonal to both of them Homework Equations I know that two vectors are orthogonal if their dot product is zero... The Attempt at a Solution I don't even know how to begin! I know the unit vector...