Questions tagged [eigenvalues-eigenvectors]
Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.
13,755
questions
0
votes
0
answers
33
views
Left eigenvector of the product of matrices
I have two matrices $A_1$ and $A_2$, and their corresponding left-eigenvectors corresponding to eigenvalue $1$ as $\mathbf w_1$ and $\mathbf w_2$.
$\mathbf w_1^TA_1=\mathbf w_1^T$ and $\mathbf w_2^...
- 33
0
votes
0
answers
17
views
Stability of generalized eigenspaces
Let $A$ be an associative $\mathbb{C}$-algebra and let $\mathfrak{r}\subset A$ be a finite dimensional Lie subalgebra of $A$ (for the commutator bracket) and assume that $\mathfrak{r}=\mathfrak{n}\...
- 1,985
0
votes
1
answer
55
views
range of $\lambda I - T$ and the null space of $\bar \lambda I - T^*$
I have a few questions about the following exercise in Stein's Real analysis.
Exercise 29
Let $T$ be a compact operator on a Hilbert Space $\mathcal H$ and assume $\lambda \neq 0$.
(a) Show that the ...
- 449
3
votes
1
answer
76
views
Why can we use the identity matrix when defining the characteristic polynomial?
We begin with $$\lambda v = Tv$$ where $\lambda$ is an eigenvalue, $v$ is an eigenvector, and $T$ is the transformation in question.
We state $$\lambda v - Tv = 0$$ We then must state $$(\lambda I - T)...
- 282
-1
votes
0
answers
27
views
Prove that the product $DAB$ of a diagonal matrix and two $PD$ matrices has at least one eigenvalue with positive real part?
Let $A$, $B$ be positive definite matrices. Let $D$ be a diagonal matrix with at least one positive entry. Can one prove that $DAB$ has at least one eigenvalue with positive real part?
- 1
6
votes
0
answers
70
views
A natural (?) proof of linear indepedence of eigenvectors of distinct eigenvalues.
Proposition. Let $T\colon V \to V$ be a linear operator. If $v_1, v_2, \ldots, v_m$ are eigenvectors of $T$ that belong to distinct eigenvalues, then they are linearly independent.
The usual proofs ...
- 101
0
votes
0
answers
28
views
Diagonalizing the sum of block matrices
Consider the matrix
$$
A=\left[\begin{array}{cc}
H_{1} & 0\\
0 & H_{2}
\end{array}\right]-\left[\begin{array}{cc}
R_{1} & R_{2}\\
R_{1} & R_{2}
\end{array}\right]
$$
where $H_1$ and $...
- 1,129
2
votes
0
answers
44
views
Density estimation with Orthogonal series
I have found this problem from the article http://www.yaroslavvb.com/papers/watson-density.pdf. In this article the probability density function has considered as
\begin{equation}
f(x) = \sum_{m=0}^{...
- 49
0
votes
0
answers
19
views
A criteron for vanishing of Lie algebra cohomology
I am reading a criterion of Professor Serre for the vanishing of cohomology of Lie algebras in his paper "Sur les groupes de congruence des variétés abéliennes II". To prove this criterion, ...
- 314
0
votes
0
answers
49
views
On the linear operator $T(M)=AM-MA$ [duplicate]
From Artin's Algebra (2nd edition)
Exercise 5.2.3 Let $A$ be an $n\times n$ complex matrix.
(a) Consider the linear operator $T$ defined on the space $\mathbb{C}^{n\times n}$ of all complex $n\times ...
- 99
0
votes
1
answer
45
views
Lemma 6.2. in Brezis' Functional Analysis
Let $(E, |\cdot|)$ be a real Banach space. For a linear map $T: E \to E$, we denote by $N(T)$ its kernel and $R(T)$ its range. Let $I: E \to E$ be the identity map.
I'm trying to prove Lemma 6.2. in ...
- 4,689
0
votes
1
answer
34
views
Do linear operators on zero vector always equal to zero?
I'm looking at a mathematical induction proof on why the eigenvectors of a linear operator T are linearly independent to each other.
So far I got to the point where for the linear combination of the ...
4
votes
1
answer
95
views
Generalization of Eigenspaces to Vector Bundles
Suppose we have a vector bundle $E$ over a smooth manifold $M$, and let $F$ be a vector bundle endomorphism $E$.
Is there anyway to now generalize the theory of eigenspaces in linear algebra to vector ...
- 1,908
1
vote
0
answers
46
views
Finding eigenvalues of $D + \alpha v v^*{}$ with $D$ being a diagonal matrix
I know how to find eigenvalues of an $n \times n$ Hermitian matrix
$$H = I + \alpha vv^{*},$$
where $\alpha \in \mathbb{R}$, $v \in \mathbb{C}^n$ is a unit length vector. We can write
\begin{align}
(I ...
- 231
0
votes
0
answers
19
views
Order of the vectors in principal component analysis
For a given matrix $A$, the Principle Component Analysis (PCA) is done by finding the eigenvalues/eigenvectors of the covariance matrix associated with $A$. However, the entries of the covariance ...
- 453
1
vote
1
answer
55
views
Maximal eigenvalue of Hermitian matrix lies on the diagonal of $A$
Let $A$ be Hermitian. Assume that the eigenvalues of $A$ are increasing ordered and that $a_{ii} = \lambda_n$. We have that $$\sum_{j=1}^{m}a_{ii} \geq \sum_{j=1}^{m}\lambda_j$$ and $$\sum_{j=1}^{m}a_{...
- 79
0
votes
0
answers
34
views
Is it possible to show that the eigenbasis is the most 'optimal' basis for a given linear transformation (if it exists)? How would you measure this?
Given a linear transformation, a coordinate transformation to the eigenbasis (if it exists) seems the most optimal and easiest to calculate. An eigenvector expressed in other coordinates always seems ...
- 519
0
votes
1
answer
35
views
Algorithm for diagonalization of a bilinear form
I've got introduced to diagonalization for linear transformation ( eigenvalues, eigenvectors, algebraic/geometric multiplicity... ) and everything was quite clear, you've got the expression $Av = \...
0
votes
1
answer
33
views
Estimation consistency of the largest eigenvalue and its corresponding eigenvector of the estimated matrix
I am estimating a statistical model. Let $\Sigma=\beta\beta^\top + D$, where $\beta\in\mathbb{R}^p$ and $D$ is a $p\times p$ diagonal matrix. Denote $\widehat{\Sigma}$ as a consistent estimator for $\...
- 51
0
votes
2
answers
15
views
Eigenvalues of skew-symmetric matrices from 2D random points
I am generating $n$ random points in two dimensions $(x_i, y_i)$. Then I form this skew-symmetric matrix
$$ M_{ij} = \begin{cases} x_i y_j - x_j y_i & \text{if } i < j \\ 0 & \text{if } i=...
- 453
0
votes
1
answer
27
views
Proof of non-singularity of $F'(y)$ when $\mu$ is a simple eigenvalue of the real symmetric matrix $A$
Consider the nonlinear system of equations：
$$
F(x,\mu) = \begin{bmatrix}
Ax - \mu x \\
\frac12 x^T x - 1
\end{bmatrix}
$$
where $A$ is a real symmetric matrix and $\mu$ is a simple eigenvalue.
To ...
- 31
0
votes
0
answers
31
views
Has it been proven that the Cheeger constant is attainable on surface?
Recently, I have been studying the monotonicity of the Cheeger constant under Ricci flow on surfaces. In fact, I want to use the monotonicity to prove the convergence of Ricci flow on $S^2$, which ...
- 5,495
2
votes
1
answer
49
views
How to compute quickly the Jordan normal form of a product of the form $B A \operatorname{adj} B$? [duplicate]
Compute the Jordan normal form of the product $$BA\operatorname{adj}(B),$$ where
$$A = \begin{pmatrix}
1 & 3 & -1 \\
-1 & 4 & 0 \\
0 & -1 & 4 \\
\end{pmatrix}, \qquad
B = \...
- 27
1
vote
2
answers
37
views
If every eigenvector of $\Phi + \Phi^{*}$ is also an eigenvector of $\Phi - \Phi^{*}$, $\Phi$ is normal.
Let $V$ be an inner product space with finite dimension and $\Phi: V \rightarrow V$ a homomorphism. If every eigenvector of $\Phi+\Phi^{*}$ is also an eigenvector of $\Phi-\Phi^{*}$, how can we ...
- 147
1
vote
1
answer
128
views
Linear operator $A: \mathbb R^n \to \mathbb R^n$
Linear operator $A: \mathbb R^n \to \mathbb R^n$ such as $A^3$ — projection operator. What eigenvalues could $A$ has? Is it correct that $A$ will have diagonal matrix in some $\mathbb R^n$ basis?
My ...
1
vote
0
answers
20
views
Divergence of smallest eigenvalue of a block positive semidefinite matrix
Let $X_1$ and $X_2$ are $n \times k$ and $n \times p$ matrices. Then $X=(X_1,X_2)$ is $n\times (k+p)$ matrix. Define $H= X^{\top}X$. Then
$$H = \begin{pmatrix}
X_1^{\top}X_1 & X_1^{\top}X_2\\
...
- 104
2
votes
1
answer
43
views
Show distinct eigenvectors of a Hermitian matrix are orthogonal
Let $A$ be a Hermitian matrix (i.e. equal to its conjugate transpose $A^H$) having eigenvectors $x, y$ with distinct eigenvalues $\lambda_1, \lambda_2, \lambda_1 \neq \lambda_2$. Show $x$ and $y$ are ...
- 2,183
0
votes
0
answers
19
views
Sagemath -- finding Eigenvalues of a matrix representation of a tensor
I am using sagemath to compute Einstein tensors of a non-standard spacetime. The output is something horrid and non-diagonal. I need to find the Eigenvalues of this tensor... which is represented as a ...
- 13
-1
votes
1
answer
34
views
Verify my proof that eigenvalues of a unitary matrix have absolute value 1
Show that eigenvalues of a unitary matrix have absolute value 1.
Proof below. Please verify, critique, or improve. Note: Many proofs are available; this question is to verify this proof.
Notation: $...
- 2,183
-3
votes
0
answers
33
views
Construct a permutation matrix from some eigenvectors and eigenvalues.
Given $n$ orthonormal vectors $v_1, \dots, v_n \in \mathbb R^d$, where $d > n$, there are many orthogonal matrices of size $d$ such that $v_1, \dots, v_n$ are their eigenvectors corresponding to ...
- 2,390
0
votes
1
answer
35
views
Looking for eigenvalues
I'm getting a bit confused here. I think I was told that all $n\times n$ matrices had eigenvalues, but to me the following matrix seems suspicious:
\begin{bmatrix}
4 && 5 \\
-5 &&...
- 23
1
vote
1
answer
23
views
Why if $u$ satisfies $-\Delta u+tu=f$ and $\int_{X}\nabla u\cdot\nabla v=\lambda\int_{X}uv$ has non-trivial solutions $u$ then $-\lambda\leq c$
Let $X\subset\mathbb{R}^2$ be a bounded open set with either Dirichlet $u = 0$ or Neumann boundary $\frac{\partial u}{\partial \nu} = 0$ conditions on the boundary of $\Omega$. Let $V$ be either $H^...
- 695
2
votes
2
answers
111
views
Proving that A=A(B−I) and B=B(A−I) implies A is diagonalizable.
A, B matrices nxn:
Proving that A=A(B−I) and B=B(A−I) implies A is diagonalizable.
I found that 2 and 0 are the eigenvalues of A. and then i tried to prove that rank(2I-A)=rank(A) if so the nul(A) + ...
- 31
0
votes
0
answers
23
views
$\pm 1$ matrices with constant row and column sums
I am facing matrices in which every entry is $\pm 1$ and every row and column sums to the same constant, e.g.,
$$M=\begin{pmatrix}
1 & -1 & -1 & 1 & -1\\
-1 & -...
- 45
2
votes
0
answers
45
views
Why does asymptotically stable fixed point become saddle structure in traveling wave coordinates?
Suppose the reaction-diffusion system
Suppose the dynamical system
\begin{align*}
v_t &= f(v, w) \\
w_t &= g(v, w)
\end{align*}
where $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$,
has an ...
- 77
-1
votes
0
answers
22
views
Find a basis for the orthogonal to span (V1,V2) [closed]
v1=2
1
v2=0
1
0
Find a basis for the orthogonal complement to span (v1,V2) where:
-1
votes
1
answer
45
views
Eigenvalue inequality $v^tAv \geq \lambda \|v\|_2$
Let's say you have a vector $v$ and a matrix $A$ whose minimum eigenvalue is $\lambda$. I have seen people use the following inequality, but I don't understand how they arrived at it.
$$v^t A v \geq \...
1
vote
0
answers
28
views
Spectral norm of product and spectral radius
I have been thinking about the following problem on the upper bound of the spectral norm of the product: Consider $||\cdot||$ as the spectral norm, by the definition of matrix norm we have
$$||AB||\...
- 79
2
votes
1
answer
45
views
Solving a modified matrix equation using solutions of the original equation
I am trying to find analytically the eigenmodes of the following equation for a damped system with $N$ degrees of freedom:
$$ \mathbf{M}\ddot{\mathbf{u}} + \mathbf{C} \dot{\mathbf{u}} + \mathbf{K}\...
- 51
0
votes
2
answers
78
views
Solution of equation $(1 - c)[(2 - c) ^ 2 - 9] = 0$
I was solving a characteristic equation for matrix eigenvalue problem. I had this equation, where $c$ is eigenvalue.
$$(1 - c)[(2 - c) ^ 2 - 9] = 0$$
In the book they equated both the terms to zero, ...
2
votes
1
answer
37
views
Do solutions to the matrix / vector equation $(M-k*N)u=0$ diagonalize M and N at the same time?
Let's say I am given this equation:
\begin{align}
(M-k*N) \cdot u = 0
\end{align}
for given positive symmetric real matrices $M$ and $N$ of Dimension n, and yet to be determined constant $k$ and ...
- 255
0
votes
0
answers
23
views
Localization of eigenvalues for block-tridiagonal Hermitian Toeplitz matrix made of gamma blocks
I am studying the spectrum of a particular kind of block-tridiagonal Hermitian Toeplitz matrix made of three bands $\{B,A,C\}$
$$ T_n = \begin{pmatrix}
A & C & 0 & \dots & 0\\
...
- 111
1
vote
1
answer
62
views
If $\lambda$ is eigenvalue for $A$, then $k\lambda$ is eigenvalue for $kA$
We know that
if a matrix $A$ has eigenvalues $\lambda_1,\dots,\lambda_n$, then the matrix $kA$ has eigenvalues $k\lambda_1,\dots,k\lambda_n$.
An example is the matrix
$$A=\left( \begin{matrix}1 &...
- 1,306
0
votes
0
answers
14
views
Eigendecomposition in unconstrained QP
Can someone please help me to understand the following statement?
Let $Q \in \mathbb{R}^{d \times d}$ be a positive definite matrix with eigenvalues $\lambda_{\max} = \lambda_{1} \geq \dots \geq \...
- 23
0
votes
1
answer
49
views
analysis of matrix multiplication via singular-value-decomposition? [closed]
Consider a square real matrix $\mathbf{A}$ and a real symmetric matrix of the same size $\mathbf{S}$
If I can diagonalise $\mathbf{A}$, such that $\mathbf{A}=\mathbf{B}\mathbf{D}\mathbf{B}^{-1}$, then ...
- 127
2
votes
2
answers
97
views
Is "if $A v \in \text{Vect}(v)$ then $v$ is an eigenvector" correct? [closed]
I am very sorry if it's a stupid question. In my homework, to prove that a vector $v_1 \neq 0$ is an eigenvector of a matrix $A$ with nonzero eigenvalue, I've proved:
$A v_1 \neq 0$, and
$A v_1 \in \...
- 253
0
votes
1
answer
50
views
If I have complete set of vectors that are orthogonal with respect to a matrix, does that mean they are eigenvectors of that matrix?
I have a symmetric full rank matrix M, and a set of vectors $v_i$ that span the whole space. I assume those vectors to be orthogonal (they form an orthogonal set).
If two of those vectors $v_i$ and $...
- 255
0
votes
0
answers
28
views
"Height" of the positive eigenvector of a non negative irreducible symmetric stochastic banded Toeplitz matrix
Let $ (a_n) $ be a sequence of non-negative real numbers such that $\sum_{i = 0 }^\infty a_i = 1$. We assume that only a finite numbers of the $a_i$s are non zero. Let $M_d$ be the $d \times d$ ...
- 11
0
votes
1
answer
41
views
Solving a differential equation with given eigenvalues and eigenvectors
I am currently working on a differential equation problem and could use some help in solving it. The problem is to find the general solution of the differential equation:
$$
\mathbf{x}^{\prime}=A \...
- 167
3
votes
1
answer
52
views
Heat Equation Separation of Variables -> Deriving X and Y Equations from boundary conditions
I am trying to practice for an exam that will have material involving separation of variables with the heat equation. While searching for practice problems on the internet, I found this pdf of ...
- 65