1
vote
1answer
37 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
0
votes
1answer
47 views

Do T and T* have the same eigenvalues with the same algebraic multiplicity?

I know that the eigenvalues of T* are the conjugates of T's eigenvalues , but how can I see each eigenvalue of T and it's conjugate , the eigenvalue of T*, have the same algebraic multiplicity?
0
votes
3answers
51 views

find the vector $(x,y,z) \in \mathbb{R}^3$ and the constants $\lambda \in \mathbb{R} $ such that $T(x,y,z) = (\lambda x, \lambda y, \lambda z )$

Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ defined by : $$T(x,y,z) = (x-y+4z,3x+2y-z,2x+y-z)$$ How can i find the vector $(x,y,z) \in \mathbb{R}^3$ and the constants $\lambda \in \mathbb{R}$ ...
0
votes
0answers
18 views

PCA - How to calculate the scores

I'm currently learning Principle component analysis and I have, so far calculated the Eigen values and vectors. Assume that I have the following: $$ E = \begin{pmatrix} 1 & 2\\ 3& 4 ...
1
vote
3answers
45 views

Help on finding eigenvalues of transformation on matrices

T is linear transformation working on 2x2 matrices: T(A) = $\begin{bmatrix}1 & 1\\1 &1\end{bmatrix}$ A as far as I see only 0 is an eigen value but someone told me 2 is eigen value too and ...
0
votes
2answers
30 views

Diagonalization of a strange transformation

Let be $V$ a vector space on $\mathbb C$ and $\dim V=4$ and let be $f \in \operatorname{End}(V)$ such that $\operatorname{Im}(f^2+a \cdot \operatorname{id}) \subset \ker(f+id)$ where $a=\det f$, ...
1
vote
1answer
48 views

I need hints on showing a matrix with certain properties defines a special transformation

Given the matrix $$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$ with integer coefficients, rational eigenvalue, and determinant $1$, show $A$ acts as a shearing along its eigenvector. Here is ...
1
vote
3answers
26 views

Linear transformations and eigenvalues [duplicate]

Let $T: \mathbb C^n \rightarrow \mathbb C^n$ be linear. Let $\beta$ and $\gamma$ be any two ordered bases. Prove that the eigenvalues of $[T]_\beta$ and $[T]_\gamma$ are the same. Can anyone provide ...
0
votes
0answers
57 views

Linear independent eigenvectors and eigenvalues

I have T as a linear transformation from V to V over the field F, V has dimension n. T has the maximum number n distinct eigenvalues, then show that there exists a basis of V consisting of ...
1
vote
1answer
170 views

Finding Eigenvalues of Block Matrix

I have a block matrix of size $3N \times 3N$ of the form: $B = \left[\begin{array}{cccc} A & C & \dots & C\\ \vdots & A & \dots & C\\ C & \vdots & \ddots & ...
1
vote
0answers
97 views

Infinite dimensional vector space eigenvectors eigenvalues and representation

We can express linear transformations with their eigenvectors and eigenvalues in finite vector spaces if they are diagonalizable. even if they are not diagonalizable we can express them via Jordan ...
2
votes
1answer
195 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
0
votes
2answers
174 views

Long-term behaviour of a linear transformation (Is the domain eventually mapped onto the dominant eigenspace?)

As far as its coordinate representation is concerned, the domain of a linear transformation will eventually (i.e. after infinitely many iterations of the transformation) be mapped onto the dominant ...
1
vote
1answer
154 views

Calibration of an eye tracking device: transformation from known gaze points

I am creating a calibration system for an eye tracking device. This calibration involves having the user look at five points on a screen. The eye tracker then reports where it believes the user was ...
7
votes
6answers
1k 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 ...