0
votes
0answers
19 views

Find real solution for an inhomogene system

I have an inhomogene differential equation system $\begin{pmatrix}\dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}-1 & 3 \\ -3 & -1\end{pmatrix} \begin{pmatrix}x_1 \\ x_2\end{pmatrix} + ...
3
votes
0answers
60 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
2
votes
3answers
324 views

Suppose A has eigenvalues 1,2, 4.

a) What is the trace of $A^2$ b) What is the determinant of $(A^{-1})^T$ I need someone to check my answers and correct me, am especially not sure about part a), help me me out; for a), I did--- ...
0
votes
2answers
31 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
0
votes
1answer
40 views

$T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ and $ <A,B> = Tr(AB^t)$

Let $V = M_{n \times n}(R)$ with the inner product $ <A,B> = Tr(AB^t)$, and $T$ the linear operator given by $T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ . How can i ...
0
votes
1answer
14 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 ...
1
vote
1answer
18 views

linear combination vectors into one vector

write $x(a_1,a_2,a_3)+y(b_1,b_2,b_3)+z(c_1,c_2,c_3)$ as $Y(a_1,a_2,a_3,$ $b_1,b_2,b_3, $ $c_1,c_2,c_3)$ ^as a 3x3 matrix for a suitable Y?
2
votes
1answer
27 views

Dimension of vector space (count 0 or not)

Do you count 0 in the dimension of a vector space? Eg. If $V_\lambda$ is the eigenspace of a certain function $f$, which has eigenvectors corresponding to $\lambda$ of $v_1, v_2, v_3$ then the basis ...
0
votes
3answers
63 views

Find eigenvector of the linear operator

Task is to find an eigenvector of the following linear operator: $f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$. I know how to find eigenvectors ...
2
votes
1answer
23 views

Invariant Subspaces - Eigenvectors

I have the following function: $\rho : C_4 \rightarrow GL_2(\Bbb C)$ $\rho(e) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} $ $\rho(x) = \begin{pmatrix} 0 & i \\ i ...
1
vote
1answer
88 views

$\mathrm{col}(AB) = \mathrm{col}(B)$

Let $A$ be a real invertible $n\times n$ matrix, and $B$ a real $n\times m$ matrix. $\mathrm{col}(.)$ denotes the column space of a matrix. What are the conditions for $\mathrm{col}(AB) = ...
0
votes
2answers
115 views

Vector Space-Linear Transformation (Multiple Choice)

I was trying to solve the following problem from a competitive exam paper. Let $A$ be a nonzero linear transformation on a real vector space $V$ of dimension $n$. Let the subspace $V_0 \subset V$ be ...
1
vote
1answer
56 views

Why is the intersection of spans zero?

As a part of a larger proof, my text claims that if $$A\begin{bmatrix}u_1&u_2\\ \end{bmatrix}=\begin{bmatrix}u_1&u_2\\ \end{bmatrix}\begin{bmatrix}\lambda&1\\0&\lambda\\ ...
1
vote
1answer
49 views

Existence of invariant subspaces of $\mathbb C^n$ and $\mathbb R^n$

Exercise $6.15$ from Dym's Linear Algebra in Action: Let $A \in \mathbb R^{n\times n}$ and suppose that $n \geq 2$. Show that: $(1)$ There exists a one-dimensional subspace U of $\mathbb ...
0
votes
1answer
62 views

Invariant subspaces for endomorphisms with associated Jordan matrices

I would like to know which are the invariant subspaces for the endomorphisms $f1$, $f2$, $f3$, $f4$, $f5$ from vector space $V$ that have the next associated Jordan matrices: $J1 = \left( ...
0
votes
2answers
32 views

Proving eigenvalue $\lambda$ is a root for an annihilating polynomial $p(t)$

I have the next statement I need to prove: Let $\lambda$ be an eigenvalue of a endomorphism $f$ and $p(t)$ an annihilating polynomial of $f$. Prove that $\lambda$ is a root for $p(t)$. Thank you ...
0
votes
1answer
49 views

Proving subspaces are invariant for different statements [closed]

I would like to know how to prove the next statements regarding invariant subspaces: Statement 1: $f$ and $g$ are endomorphisms from a vector space $V$. If $f$ and $g$ commute, then subspaces ...
2
votes
0answers
101 views

A basis for the column space of a real matrix

Let $A$ be a real square matrix, and let its column space be $$\mathrm{col}(A)=\{y\in\mathbb{C}^n:y=Ax\text{ for some } x\in\mathbb{C}^n\}.$$ Under what conditions is $\mathrm{col}(A)$ spanned by ...
1
vote
1answer
22 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
0
votes
1answer
42 views

Invariant subspaces using matrix of linear operator

I am attempting the following problem but stuck at some parts: How does one find the (2 dimensional) subspaces that are invariant under $A$ for $$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 &2 ...
0
votes
3answers
45 views

A question about eigenvectors.

Let $T\in L(V,V)$, and let $\{v_1,v_2,\dots,v_n\}$ be a basis of $V$ consisting of eigenvectors of $T$, belonging to eigenvalues $a_1,a_2,\dots,a_n$ respectively. Then $Tv_i=a_iv_i$. Prove that ...
0
votes
1answer
56 views

Stochastic matrix problem

A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to 1. Let $A$ be any (general) 2x2 stochastic matrix. a) Show that one of the ...
0
votes
0answers
38 views

Rank of a large matrix

Suppose i want to calculate rank of a large $N\times N$ matrix having only $0$ and $1$ which is represented by $M$ segments each segment here is depicting that for a segment [LEFT,RIGHT] the row of ...
0
votes
2answers
35 views

A question about eigenvalues

Let $v=\begin{pmatrix}v_1\\v_2\\v_3\\v_4\end{pmatrix}$ be a nonzero column vector in $\Bbb R^4$ and let $A=vv^T$. Find the eigenvalues of $A$ There must be a easier way rather than calculate it ...
1
vote
4answers
59 views

How to find vector that is orthonormal on other.

I have to generate a Q matrix for Schur decomposition and I have the first column, let's it is the following: \begin{bmatrix}1/√3\\1/√3\\1/√3 \end{bmatrix} Now I need to find the second column that ...
1
vote
1answer
148 views

Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
4
votes
1answer
125 views

All Invariant Subspaces of a Linear Transformation

I got this problem: Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that all it's eigenvalues are 1, 2 and 3 and the corresponding eigenvectors are $v_1, v_2$ and $v_3$ ...
0
votes
1answer
64 views

Variations in math to implement three-dimensional space?

Backstory: So I was researching topics, and found that 3-D game programming often markets itself with linear algebra. As a philosopher of math I decided to dig further into this and determine if ...
8
votes
1answer
100 views

Prove the operators $T+U$ and $U$ have the same eigenvalues where $T$ is nilpotent

Let $V$ be an $n$-dimensional vector space on $\mathbb{C}$, and $T$ a nilpotent operator on $V$. Let $U$ be in $L(V)$ s.t. $UT = TU$. Prove that the operators $T+U$ and $U$ have the same eigenvalues. ...
2
votes
3answers
153 views

Eigenvectors are linearly independent?

Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent. Could someone give me a geometric interpretation of the theorem? Thanks!
0
votes
1answer
65 views

Intuition behind independence of eigenvectors?

Theorem 6.14: Eigenvectors corresponding to distinct eigenvalues of A are linearly independent. My prof already gave us a proof of the theorem, so I'm not looking for another one. Could someone ...
0
votes
1answer
66 views

Eigenspace and polynomials?

My prof introduced us to eigenvectors and eigenvalues today. He then gave us the following theorem: Theorem 6.6: Let $A$ be a square matrix, let $\gamma$ be an eigenvalue of $A$ with multiplicity ...
1
vote
1answer
37 views

Eigenvector and Its Span

Let $V$ be a vector space over the field $F$ and let $T$ be a linear transformation from $V$ to $V$. Let $v\in V$ such that $v\neq 0$, let $W=span\{v\}$. Prove that if $T(W) \subset W$, then $v$ is an ...
3
votes
2answers
117 views

Can you factor out vectors?

My prof introduced eigenvalues to us today: Let $A$ be an $n \times n$ matrix. If there a scalar $\lambda$ and an $n-1$ non-zero column vector $u$, then $$Au = \lambda u$$ then $\lambda$ ...
0
votes
1answer
32 views

How to show that the limit of sequence of eigenvectors (same eigenvalue) is also an eigenvector?

Let $H$ be a continuous Hermitian operator on an infinite dimensional Hilbert space. Also, let $f_n$ be a sequence approaching $f$ as $n\to\infty$, where each $f_n$ is an eigenvector of the same ...
3
votes
1answer
189 views

Disjoint Gershgorin disks $\Rightarrow$ each contains exactly one eigenvalue

It is an exercise in Peter Lax's book Linear Algebra that if all the Gershgorin disks $$D_i := \{z\in \mathbb{C} : |a_{ii} - z| \leq \sum_{i \neq j} |a_{ij}|\}$$ are disjoint, then each disk must ...
1
vote
1answer
23 views

Layering of eigenvalues

If $\tau: V \to V$ is a linear transformation on a finite-dimensional real vector space with eigenvalues $a_1, \dotsc, a_n$, in ascending order, and $P:V\to V$ is orthogonal projection onto $W ...
5
votes
1answer
219 views

What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
0
votes
0answers
134 views

Has this Principal Component Analysis (PCA) been done correctly?

I have a set of 3D data points, indicated by the blue color in the picture below. I then project them onto the x-y plane, i.e. setting z values of all the points to 0, shown by the yellow color ...
1
vote
0answers
33 views

Why is Lax's statement of the minimax principle stated only for real vector spaces?

I am reading Peter Lax's Linear Algebra. On page 116 he states the minimax principle as follows: for a real symmetric operator $H$ on an $n$-dimensional real vector space, suppose we label the ...
2
votes
0answers
65 views

What is the importance of eigenvalues in mathematics ? In the real world, where we use them? [duplicate]

What is the importance of eigenvalues in mathematics? In the real world, where do we use these eigenvalues?
2
votes
1answer
73 views

Find Eigenvectors of a homomorphism over polynomial vector spaces

Let $\mathbb{R}[x]$ be the real-valued vector space of polynomials with real-valued coefficients and $F: \mathbb{R}[x]\rightarrow\mathbb{R}[x]$ be a homomorphism defined as $$ ...
3
votes
2answers
278 views

Prove $T$ has at most two distinct eigenvalues

The question is from Axler's Linear Algebra text. The $\mathcal{L}(V)$ stands for the space of linear operators on the vector space $V$. Suppose that V is a complex vector space with dim ...
1
vote
2answers
315 views

Matrix representation of the adjoint of an operator, the same as the complex conjugate of the transpose of that operator?

Since I'm not taking summer classes I decided to do some self learning on more advanced mathematics, and I've found myself stuck on this problem: I have to show that for any operator $\hat{A}$ the ...
5
votes
0answers
184 views

Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?

An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
4
votes
1answer
105 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
294 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 ...
3
votes
2answers
60 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
100 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 ...
2
votes
1answer
262 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), ...