Tagged Questions

4
votes
1answer
39 views

To prove that the dimension of $V$ is $d_1^2 + \ldots + d_k^2$

Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $$(x - c_1)^{d_1} \cdots (x - c_k)^{d_k} , $$ where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n \times n$ ...
3
votes
2answers
54 views

Proof of the linear independence of the generalized eigenvectors of a square matrix

I'm currently stuck on this problem: Let $V$ be a finite dimensional vector space. If $S: V\rightarrow V$ and $T: V\rightarrow V$ are linear maps and $ST=TS$, prove every eigenvalue of $ST$ is a ...
2
votes
2answers
37 views

Generalised eigenvalue is eigenvalue if it is in the field

I would like to prove the following assertion: Let $\mathscr{F}$ be a field and $\mathscr{\phi}$ be an $\mathscr{F}$-linear endomorphism of a finite dimensional $\mathscr{F}$-vector space ...
3
votes
1answer
46 views

Eigenvalues and Eigenvectors Diagonilization

Let $ A=\begin{bmatrix} -7 & -1 \\ 12 & 0 \\ \end{bmatrix} $ . Find a matrix $ P $ and a diagonal matrix $D$ such that $PDP^{-1} = A$. Ok so the first thing I need to look ...
1
vote
3answers
43 views

Differentiation operator and eigenvalues

Let $V = \{p(x) \in F[x] \ | \ \deg(p(x)) \le n\}$. Let $T : V \to V$ be given by differentiation, in essence $$T(p(x)) = p'(x)$$ It seems to me that the only eigenvalue that can exist is $\lambda ...
3
votes
2answers
111 views

Matrix proof using norms

I have a linear algebra question I need help with. Let $A$ be an $m\times m$ matrix with $\|A\|_2 < 1$ where $\|A\|_2$ is the $2$-norm of $A$. Show that $I - A$ is invertible where $I$ is the ...
1
vote
1answer
66 views

Dependency of vectors in eigenspace corresponding with eigenvalue zero

The eigenspace corresponding with the eigenvalue zero is the same as the null space of the original matrix. All vectors in the null space are linearly independent so the eigenvectors of zero are also ...
1
vote
2answers
52 views

Diagonalizablility of $T$

Let $M_2(\mathbb R)$ denotes the set of $2\times2$ real matrices. Let $A\in M_2(\mathbb R)$ be of trace $2$ and determinant $-3$. Identifying $M_2(\mathbb R)$ with $\mathbb R^4$, consider the linear ...
1
vote
2answers
173 views

SVD on columns of a rotation matrix

Suppose a matrix $A\in\mathbb{R}^{n\times m}$ is given, $n>m$, with columns being subset to those of an rotation matrix (i.e., matrix with with orthonormal columns). Is it true that the sigular ...
1
vote
1answer
38 views

Small perturbations

Background: Let $x_1,\ldots,x_n$ be the variables satisfying the equations of motion $\ddot{x_i}=f_i(x_1,\ldots,x_n)$ for $i=1,\ldots,n$ We introduce a small perturbation such that $x_i(t)=x_i^0 ...
0
votes
0answers
50 views

Finding point distribution by eigen vectors

First of all I want to tell that my mathematics is poor, so I can’t use correct terms. Sorry for that. I have a point data set. This data represents some cylindrical objects surfaces (not exactly ...
3
votes
2answers
131 views

$2\times 2 $ matrices over $\mathbb{C}$ that satisfy $A^3=A$

Let $A$ be a $2\times 2$ matrix with complex entries. What would be the number of $2\times 2$ matrices $A$ that satisfies $A^{3} = A$. Question was are they infinite? If it is $3\times 3$ matrix then ...
1
vote
1answer
179 views

Spectrum shift except for zero eigenvalue

Suppose a square, real and symmetric matrix $G\in\mathbb{R}^{n\times n}$ is given, and it is known to have one zero eigenvalue associated with all ones eingenvector, $1_n$. I'm aware that the ...