1
vote
1answer
20 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
19 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
42 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
39 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
45 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
101 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
70 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
51 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
87 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
117 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
59 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
54 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
36 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
81 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
31 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
162 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
21 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 ...
3
votes
1answer
167 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
122 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
31 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
62 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
66 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
231 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
180 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
169 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
87 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
249 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
55 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
88 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
239 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), ...
1
vote
3answers
111 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
210 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
146 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
63 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
325 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
50 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
63 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
164 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 ...
5
votes
4answers
990 views

Dot product of two vectors without a common origin

Given two unit vectors $v_1, v_2\in R^n$, their dot product is defined as $v_1^Tv_2=\|v_1\|\cdot\|v_2\|\cdot\cos(\alpha)=\cos(\alpha)$. Now, suppose the vectors are in a relation $v_2=v_1+a\cdot1_n$, ...
1
vote
1answer
305 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 ...
0
votes
2answers
210 views

What is the use of a transpose of a matrix in an equation and how to solve one?

I have the following equation to solve, $$g(x) = x^t W_i x + {W_i}^t x + v_{i_0}$$ In this equation why the need to use a $x^t$ and $x$? I feel $x$ and transpose of it both are the same ($x$ is a row ...
0
votes
1answer
221 views

Eigenvector corresponding to zero eigenvalue / identical eigenvalues, not-identical eigenvectors

Assume symmetricmatrix $B\in\mathbb{R}^{n\times n}$ is given, and a transformation $$A=JBJ,$$ where $J=I - \frac{1}{n}1_n1_n^T$ and $I$ denoting the identity matrix, hence centering its rows and ...