Orthogonal Definition and 584 Threads

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...

Homework Statement Let P\inL(V). If P^2=P, and llPvll<=llvll, prove that P is an orthogonal projection.Homework Equations The Attempt at a Solution I think that regarding llPvll<=llvll is redundant. For example, consider P^2=P and let v be a vector in V. Doesn't P^2=P kind of give it away by...

Aren't all projections orthogonal projections? What I mean is that let's say there is a vector in 3d space and it gets projected to 2d space. So [1 2 3]--->[1 2 0] Within the null space is [0 0 3], which is perpendicular to every vector in the x-y plane, not to mention the inner product of...

I know what orthogonal means (well, I know orthogonal vectors are perpendicular to each other) but how can this be applied to a wavefunction? Thanks!

Homework Statement Let W be the subspace spanned by the given column vectors. Find a basis for W perp. w1= [2 -1 6 3] w2 = [-1 2 -3 -2] w3 = [2 5 6 1] (these should actually be written as column vectors. Homework Equations The Attempt at a Solution So, I...

Hi, I was just reading about Orthogonal complements. I managed to prove that if V was a vector space, and W was a subspace of V, then it implied that the orthogonal complement of W was also a subspace of V. I then proved that the intersection of W and its orthogonal complement equals 0...

Homework Statement I would like to show that SO3 does not contain any subgroups that are isomorphic to SO2 X SO2. Homework Equations I know that any finite subgroup of SO3 must be isomorphic to a cyclic group, a dihedral group, or the group of rotational symmetries of the tetrahedron...

I need some direction. Suppose that{u1,u2,...,um} are non-zero pairwise orthogonal vectors (i.e., uj.ui=0 if i doesnt=j) of a subspace W of dimension n. Show that m<=n.

I need some direction. I don't have a clue where to start. Suppose that{u1,u2,...,um} are non-zero pairwise orthogonal vectors (i.e., uj.ui=0 if i doesnt=j) of a subspace W of dimension n. Show that m<=n.

Show that the parabolic coordinates (u,v,\phi) defined by x=uv \cos{\phi} , y=uv \sin{\phi} , z=\frac{1}{2}(u^2-v^2) now I am a bit uneasy here because to do this i first need to find the basis vector right? so if i try and rearrange for u say and then normalise to 1 that will give me...

Homework Statement Let P2 denote the space of polynomials in k[x] and degree < or = 2. Let f, g exist in P2 such that f(x) = a2x^2 + a1x + a0 g(x) = b2x^2 + b1x + b0 Define <f, g> = a0b0 + a1b1 + a2b2 Let f1, f2, f3, f4 be given as below f1 = x^2 + 3 f2 = 1 - x f3 = 2x^2 + x + 1 f4 = x +...

Homework Statement Find an orthogonal basis for the nullspace of the matrix [2 -2 14] [0 3 -7] [0 0 2] Homework Equations The Attempt at a Solution The nullspace is x = [0, 0, 0], so what is the orthogonal basis? It can be anything can't it?

Given a = ( 1, -2, 1), b= ( 0, 1, 2), and c = (-5, -2, 1) determine if {a, b, c} is an orthogonal set. Show support for your answer. I know that if the dot product of every combination equals zero, the set is orthogonal. No problems here. I do that, and they all equal zero. a dot b: 1 * 0 +...

I have an assignment question to find an equation of the orthogonal projection onto the XY plane of the curve of intersection of twp particular functions. If some one knows of a good web page that might explain this to me I would be greatly appreciate it. regards Brendan

V is a space of inner muliplication. W1 and W2 are two subspaces of V, so dimW1<dimW2 prove that there is a vector 0\neq v\epsilon W_2 which is orthogonal to W1 ??

W is sub space of R^4 which is defined as http://img21.imageshack.us/img21/1849/63042233.th.gif find the system that defines the complements W^\perp of W i have solved the given system and i got one vector (-1,1,0,0) so its complement must be of R^3 and each one of the complements...

Let O(n,F_q) be the orthogonal group over finite field F_q. The question is how to calculate the order of the group. The answer is given in http://en.wikipedia.org/wiki/Orthogonal_group#Over_finite_fields". This seems to be a standard result, but I could not find a proof for this in the basic...

Imagine there is a molecule which consists of several atoms, and for each atom there is an effective orbital, phi_i, which are not orthogonal. Now we want to construct from them a set of orthogonal orbitals, psi_i. Of course there are many ways to do this. Let W be the matrix that realizes our...

Homework Statement Find all vectors that are perpendicular to (1,4,4,1) and (2,9,8,2) The Attempt at a Solution Create matrix A = [[1,4,4,1],[2,9,8,2]] Set Ax = 0 Reduce by Gauss elimination Produces basis of (-4,0,1,0) and (-1,0,0,1) I don't know what the correct solution to...

Homework Statement Consider a normal operator A If Rperpendiculara1 is the orthogonal complement to the subspace of eigenvectors of A with eigenvalue a1, show that if y exists in all Rperpendiculara1 then Ay exists in all Rperpendiculara1 The Attempt at a Solution This could be answered very...

I'm wondering about the action of the complex (special) orthogonal group on \mathbb{C}^n. In the real case, we can use a (real) orthogonal matrix to rotate any (real) vector into some standard vector, say (a,0,0,...,0), where a>0 is equal to the norm of the vector. In other words, the action...

Homework Statement If w is orthogonal to u and v, then show that w is also orthogonal to span ( u , v ) Homework Equations two orthogonal vectors have a dot product equalling zero The Attempt at a Solution I can see this geometrically in my mind, and I know that w . u = 0 and w ...

Homework Statement 1. If I got a square orthogonal matrix, then if I make up a new matrix from that by rearranging its rows, then will it also be orthogonal? 2. True/false: a square matrix is orthogonal if and only if its determinant is equal to + or - 1 Homework Equations no...

Hi there! In order to proof the orthogonal condition aijaik=\delta_{jk} j,k=1,2,3 I write the invariance of the length of a vector in two coordinate systems: x'ix'i=xixi Using the linear transformation: x'i=ai1xi1+ai2xi2+ai3xi3 the first term becomes: aijaikxjxk My question is: why...

Homework Statement I don't understand the difference between an orthogonal complement and it's basis. In this problem: W = [x,y,z]: 2x-y+3z=0 Find w's orthogonal complement and the basis for the orthogonal complement. The Attempt at a Solution I did a quick reduced row echelon to [2,-1,3]...

Homework Statement Prove that an orthogonal transformation T in Rm has 1 as an eigenvalue if the determinant of T equals 1 and m is odd. What can you say if m is even? The attempt at a solution I know that I can write Rm as the direct sum of irreducible invariant subspaces W1, W2, ..., Ws...

Greetings, I'm a little stuck on this algebra question Find 2 unit vectors orthogonal to v1 = <3,1,1,-1>, v2 = <-1,2,2,0> and v3 = <1,0,2,-1> I know that this means that if I let z be the vector orthogonal to these 3 then, z.v1 = z.v2= z.v3 = 0 And that my two vectors is likely +/-...

Homework Statement http://img55.imageshack.us/img55/8494/67023925dy7.png The Attempt at a Solution All functions orthogonal to 1 result in the fact that: \int_a^b f(t)\ \mbox{d}t =0 Now the extra condition is that f must be continous. (because of the intersection). But...

Homework Statement Consider the vector space \Renxn over \Re, let S denote the subspace of symmetric matrices, and R denote the subspace of skew-symmetric matrices. For matrices X,Y\in\Renxn define their inner product by <X,Y>=Tr(XTY). Show that, with respect to this inner product, R=S\bot...

I'm sure there's a simple way of doing this but I just can't think of it. In R^n, given k<n linearly independent vectors, how do you find a vector that is orthogonal to all given vectors. I know cross product works for R^3, but what about R^n?

Homework Statement Given vectors a=(1,-1,0,2,1) b=(3,1,-2,-1,0) and c=(1,5,2,4,-4), which are mutually orthogonal, find a system of linear equations that a vector x must satisfy so it is orthogonal to a, b, and c.Homework Equations None I think. The Attempt at a Solution This is part A of...

I'm curious as to how the following proof is verified. I have toiled over this thing for quite a while, but haven't made any progress. I don't need a step-by-step solution, but I would appreciate any help getting it started: Given the following: Q is an orthogonal tensor e1 is a unit...

hi! I'm new to the forums, and had a question that was more calculus-related than physics. i saw another post similar to this one, but it was incomplete and i couldn't get the answer with the information on it, any chance someone could help me out? The question is: "Find a unit vector with a...