Orthogonal Definition and 584 Threads

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}.

This start out as homework but my question is not about helping me solving the problem but instead I get conflicting answers depend on what way I approach the problem and no way to resolve. I know the answer. I am not going to even present the original question, instead just the part that I have...

Homework Statement A= [1 -1 0] [-1 2 -1] [0 -1 1] find orthogonal matrix P and diagonal matrix D such that P' A P = D Homework Equations The Attempt at a Solution i got eigenvalues are 0, 1, 3 which make D=[0 0 0; 0 1 0; 0 0 3] how to find P. because in...

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...

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.

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?

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!

(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...

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...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 ¡¡ *...

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

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...