0
votes
1answer
15 views

Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
3
votes
2answers
71 views

Axis of rotation of composition of rotations (Artin's Algebra)

Say $R_1$, $R_2$ are rotations in $\mathbb{R}^3$ with axes and angles $(v_1,\theta_1), (v_2,\theta_2)$ respectively. Since $SO_3$ is a group, we have that $R_2 \circ R_1$ is a rotation with some axis ...
0
votes
1answer
27 views

Incorporating an error ellipse from eigenvalue/vectors into 3D geometry

I have a 3D point with a covariance matrix, and an associated 3D vector that begins at the point. I would like to be able to consider alternative points for the starting position of the vector, ...
21
votes
1answer
763 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
2
votes
1answer
263 views

Eigenvalues of 3x3 Covariance Matrix, Geometric Interpretation

Problem Definition I would like to code an algorithm for decomposing a covariance matrix into its eigensolution (set of eigenvalues and corresponding eigenvectors. In my specific case I want to deal ...
2
votes
0answers
66 views

Isometry in Euclidean space, matrix

I have a few questions about a proof of this theorem: $f: U \rightarrow U$, $U$ is a Euclidean space, is an isometry $\iff$ there exists an orthonormal basis in which the matrix of $f$ (let's call it ...
0
votes
1answer
111 views

Assigning eigenvectors of a covariance matrix to the variables it was generated from

Preface: This is the follow-up question according to the insights I got from MvG on my earlier post. Think of an ellipsoid in the n-dimensional space defined by $$ell: (x-\mu)'A(x-\mu)=1.$$ Then one ...
3
votes
1answer
143 views

Correspondence between eigenvalues and eigenvectors in ellipsoids

think of an ellipsoid in the n-dimensional space defined by $$(x-\mu)'A(x-\mu)=1.$$ I was calculating the volumes of n-dimensional ellipsoids like the one from above for a while, which is ...
0
votes
2answers
59 views

Find the eigenvectors and eigenvalues of A

scale by 2 in the x direction, then scale by 2 in the y direction, then projection onto the line y = x
5
votes
2answers
691 views

Find the eigenvalues and eigenvectors of A geometrically

I am really confused with this question: Find the eigenvalues and eigenvectors of A geometrically: $$ A = \begin {pmatrix} 0 & 1 \\ 1 & 0 \end {pmatrix} $$^ reflection in the line $y=x$. ...
1
vote
2answers
91 views

Linear Algebra: Geometric means

What is the geometric means of $$M=\begin{pmatrix}\cos \theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$$ I would like to show that its eigenvector is not real.
2
votes
1answer
271 views

Obtaining Least square adjusted single line by intersecting many 3D planes

I am working with many 3D planes and looking for a Least square solution for below case. IF I am having many number of 3D planes knowing only one point and the normal vector (for eg. O1 and N1), ...
2
votes
1answer
1k views

ellipse equation from eigenvectors and eigenvalues

I have a eigenvectors d1,d2 and eigenvalues v1,v2. The eigenvectors are axes of an ellipse that surrounds data points, with center u,v, and radii size of the eigenvalues. How can I find the ellipse ...
1
vote
0answers
34 views

Position on a path

So the situation is: I have a path (which is represented as a two-dimensional array of GPS coordinates) and I have a percentage position on this path. I.e. I know that a person has walked 80% of the ...
1
vote
1answer
354 views

degenerate eigenvalues

I have a problem in understanding the exact meaning of degenerate eigenvalue. I have some database and I calculate the covariance matrix among it. the obtained eigenvalues are same ( all of them ...
7
votes
6answers
2k views

Find eigenvalues of a projection and explain what they mean

Suppose B represents the matrix of orthogonal (perpendicular) projection of $\mathbb{R}^{3}$ onto the plane $x_{2} = x_{1}$. Compute the eigenvalues and eigenvectors of B and explain their geometric ...