Tagged Questions

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1answer
51 views

Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
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1answer
39 views

Solve a System with Variable

Given these matrices, how does one find two real solutions? $dx/dt$ = $\begin{bmatrix} 3 & -5\\ 5 & 3 \end{bmatrix}x$ with $x(0) = \begin{bmatrix} 2\\ -3 \end{bmatrix}$
1
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1answer
39 views

Same eigenvalues, different eigenvectors but orthogonal

I am using a two different computational libraries to calculate the eigenvectors and eigenvalues of a symmetric matrix. The results show that the eigenvalues calculated with both libraries are exactly ...
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2answers
51 views

Common eigenvector of two linear transformation matrices

I have two linear transformation matrices \begin{pmatrix} 3 & 2 \\ -2 & 1 \end{pmatrix} and \begin{pmatrix} 1-a & -a \\ a & 1 \end{pmatrix} How to find out what the value of ...
1
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1answer
44 views

generalized eigenvector for 3x3 matrix with 1 eigenvalue, 2 eigenvectors

I am trying to find a generalized eigenvector in this problem. (I understand the general theory goes much deeper, but we are only responsible for a limited number of cases.) I have found ...
3
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2answers
74 views

Diagonalising a $2 \times 2$ and $3 \times 3$ matrix

For each of the following matrices $A$, find an invertible matrix $P$ over $C$ such that $P^{-1}AP$ is upper triangular: $$A = \begin{bmatrix}4 & 1\\-1 & 2\end{bmatrix} \quad \text{ ...
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1answer
34 views

Normal matrices with orthogonal basis

we have a theorem that says that each REAL normal matrix can be written in terms of an orthonormal basis, so that it has its eigenvalues down the diagonal and 2x2 matrices of the form $\begin{pmatrix} ...
2
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1answer
53 views

Linear Algebra - Can I use RREF to solve augmented matrix for eigenvector?

I am solving David Lay's 4th edition 7.1 number 16. So here's the problem. The original matrix: $$\begin{bmatrix}-7 & 24 \\ 24 & 7\end{bmatrix} \ ,$$ with eigenvalues $\pm 25$. I am having ...
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0answers
23 views

Deriving left eigenvector from the left eigenvector of transpose

How would I find the left eigenvector of A * A' if the left eigenvector of of A' * A is v?
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0answers
46 views

Vandermonde question

I'm studying time series analysis and in my book I came a cross with the following proof (The proof is actually the last page, but I posted as much information as possible on the problem): I have ...
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2answers
34 views

Does a constant eigenvalue of a linearly parameter-dependent matrix have a constant eigenvector?

Let $A(\alpha)=A_0+\alpha A_1$, with $A_0,A_1\in\mathbb R^{n\times n}$ and $\alpha\in\mathbb R$, such that there exists a $\lambda\in\mathbb C$ with the property that for all $\alpha$: $\det(\lambda ...
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1answer
63 views

Eigenvector with eigen value of 1

How is an eigenvector with eigen value of 1, say v, multiplied by its transpose the identity matrix? v' * v = I?
-1
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1answer
25 views

diagonalizing a matrix $A$: can $P$ be bigger than $A$?

can you have a P bigger than the original A matrix? in other words after I found the eigenvalues I then found all the eigenvectors so when I constructed the P vector turns out to be bigger than my ...
1
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2answers
59 views

Eigenvector of matrix of equal numbers

For matrix the matrix $$A = \begin{bmatrix} 3&1&1\\ 1&3&1\\ 1&1&3\\ \end{bmatrix}$$ with eigenvalues $\lambda_1=5$, $\lambda_2=2$, $\lambda_3=2$, I am trying to find the ...
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0answers
33 views

Diagonalizing the sum of a matrix and a multiple of the identity matrix

Suppose we have a matrix $A = B+\lambda I$, where $B\in \mathbb{R}^{n\times n}$, $I$ is the identity matrix and $\lambda\in \mathbb{R}$. If I know the eigenvalues and eigenvectors of $B$, what can I ...
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2answers
25 views

Matrix multiplication and eigen vectors

If $a$ is a right eigenvector of $S$ and $R^T$ with eigenvalue $1$. How would determine $a^TRSa$? Is $Sa$ simply $a$? Any hints that apply here would be greatly appreciated.
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0answers
21 views

Properties of eigenvectors

If a is a right eigenvector of Z and b is a right eigen vector of Y, is a * b' a right eigenvector of Z * Y?
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2answers
64 views

Eigenvalues and Eigenvectors of $X'X$ and $XX'$

I am trying to derive (or prove) the relationship between the eigenvalues and eigenvectors of the matrices $X'X$ and $XX'$. It is fairly intuitive that they are related but I cannot derive the ...
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0answers
20 views

Simultaneous orthogonalization for rectangular matrices?

Say that I have two linearly independent vectors in 3 dimensions, $\vec{a}_1$ and $\vec{b}_2$: $$\vec{a} = \left[a_{1},\ a_{2},\ a_{3}\right]$$ $$\vec{b} = \left[b_{1},\ b_{2},\ b_{3}\right]$$ Using ...
3
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2answers
49 views

Linear algebra, eigenvectors problem

Suppose you know that A is $2x2$ and symmetric. Assume the eigenvalues are $4$ and $7$. An eigenvector for $4$ is the vector $(3, -4)$. What is an eigenvector for $7$? So first we let ...
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0answers
40 views

Help find a proof : $ \lambda $ is $f$'s eigenvalue then $f|_{V_{\lambda}} $ has Jordan's basis

Could you help me find a fairly simple proof of the following theorem? $f: V \rightarrow V, \ \ \dim V < \infty, \ \ \lambda$ is $f$'s eigenvalue $\Rightarrow \ \ f|_{V_{\lambda}}: \ V_{\lambda} ...
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3answers
64 views

Finding eigenvectors with square root eigenvalues

I have a matrix $$\begin{bmatrix}1 &-1 &2\\2 &-2 &4\\0 &1 &1\end{bmatrix}$$ Its eigenvalues are $0$, $\sqrt{5}$ and $-\sqrt{5}$ (These are checked in MATLAB to be correct). I ...
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5answers
55 views

Having Difficulty Finding Eigenvectors

I'm having a lot of problems with the following problem in Steven J. Leon's "Linear Algebra with Applications" 8th edition. Problem 6.1.1.I asks the reader to "find the eigenvalues and the ...
0
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1answer
85 views

Finding the Matrix Power of a matrix and limit

Find the matrix power, $A^k$, of $$A=\begin{pmatrix}a & 1-a \\ b & 1-b\end{pmatrix}$$ $$D=P^{-1}AP$$ $$A^k=PD^kP^{-1}$$ I think that $$P=\begin{pmatrix}1 & \frac{a-1}{b} \\ 1 & ...
0
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1answer
42 views

Hermitian matrices [duplicate]

Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such that $\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H. How to prove ...
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3answers
31 views

Calculate eigenvectors

I am given the $2\times2$ matrix $$A = \begin {bmatrix} -2&-1 \\\\ 15&6 \ \end{bmatrix}$$ I calculated the Eigenvalues to be 3 and 1. How do I find the vectors? If I plug the value back into ...
1
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1answer
110 views

Hermitian Matrices are Diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...
2
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2answers
57 views

Dimension of the corresponding eigenspace?

I'm studying for my linear exam and would appreciate any help for this practise question: You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace? A = ...
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0answers
35 views

A Has characteristic polynomial that can be reduced to linear products $\Rightarrow$ A similar to upper triangular Matrix

Prove that if $A\in M_{n}\left(\mathbb{F}\right)$ matrix with a characteristic polynomial that can be written as a product of linear elements (?) ...
2
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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
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1answer
33 views

Convergence of eigenvalues $\lambda_{i}^k$

I've been trying to solve this for a while now but can't seem to figure it out. My initial intuition is to just calculate the limit of $\lambda_{2}^k$ using de Moivre's formula like so: $\lim_{k\to ...
2
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1answer
41 views

Finding $A^n$ in terms of $P$ and $D$ (diagonalized)

My question is regarding the last two parts. I have Found $D$ and $P$, how can I obtain $A^{200}$ and det $(A^{200})$ form $PDP^{-1}?$ Thanks!
0
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1answer
49 views

Finding eigenvalues and eigenvectors

Find the eigenvalues and eigenvectors for the matrix A = (1,2,0), (-1,-2,1), (2,4,1). I end up with the polynomial x^3-5x=0 and no exact solution.
2
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1answer
52 views

Eigenvectors and Principal component

What is the difference between eigenvectrors and principal component. I got confused about this point because some researches reported that the principal components are the same eigenvectors of ...
1
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2answers
47 views

Eigenvectors and Kernel of Matrix

I'm trying to take find the eigenvectors of the matrix $$ \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}. $$ I've found the eigenvalues of $1$ and $0$. I'm ...
2
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1answer
30 views

How do I generate my linear transformation matrix in this example?

Let $T$ be the linear transformation of space of polynomials $P^3$ given by $$T(a + bx + cx^2) = a + b(x + 1) + c(x + 1)^2$$ Find all eigenvalues and eigenvectors of $T$.
2
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2answers
32 views

The connection between the rank of a matrix and its zero-mode eigenvectors

I would be most thankful if you could help me prove that if an arbitrary n by n matrix has rank m < n, then the matrix has (n-m) linearly independent eigenvectors corresponding to the eigenvalue ...
1
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1answer
95 views

A question on Eigenvalues and Eigenvectors

Hey I need quite a bit of help, I know the question might seem easy, but I'm confused with the wording. I know that the rule is $Av = \lambda(v)$. Since there are two eigenvalues, I am assuming ...
1
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1answer
51 views

Different geometrical concepts of vectors

I'm a bit confused about the various geometric concepts of vectors. I'm mainly trying to understand if we can classify any vector into one of two categories.The first category would be free ...
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 ...
3
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2answers
63 views

Solving for eigenvectors (Introduction to Linear Algebra by Serge Lang, example 2 page 240)

In "Introduction to Linear Algebra" by Serge Lang, example 2 page 240, there is the matrix $\begin{pmatrix}1&4\\2&3\end{pmatrix}$. The characteristic polynomial is $(t-5)(t+1)$. The system to ...
2
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3answers
60 views

Why is MATLAB giving me these weird eigenvectors?

I am doing a problem which involves finding the eigenvalues and eigenvectors of the matrix $$ M=\begin{bmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0 \end{bmatrix}$$ ...
0
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0answers
35 views

Limit of $A^k b$ for $k$ large

My textbook references the fact that for any matrix $A$ and any non-zero $b$ we have $$\lim_{k\rightarrow \infty} A^k b = v$$ Where $v$ is the eigenvector of $A$ with the largest eigenvalue. The proof ...
1
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1answer
68 views

Finding the eigenvalues and the eigenvectors

The matrix: $$ A=\begin{pmatrix} 1 & -1 & 1 \\ -1 & 1 & -1 \\ -1 & 1 & -1 \\ \end{pmatrix} $$ has two real eigenvalues, one of ...
3
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1answer
84 views

Diagonalizable Matrix $A^2$

How can I find a matrix $A$ such that $A^2$ is diagonalizable but $A$ is not? I have tried many different ways, but to no avail. Is there something that I am missing in the question that gives a ...
1
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1answer
77 views

How do you find the (complex) eigenvalue and each eigenspace over C

My book only has eigenvalue and eigenspace and does not say anything about complex eigenvalue and eigenspace. $$A = \begin{pmatrix} 0&4\\-1 & 0 \end{pmatrix}\hspace{10pt}B =\begin{pmatrix} ...
0
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1answer
47 views

Bounding the smallest eigenvalue of an ergodic Markov Chain

I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
1
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1answer
101 views

An eigenvector is a non-zero vector such that…

Various sources define eigenvalues and eigenvectors in slightly different ways (context independent). For example, both of the following definitions seem not to exclude the zero-vector as an ...
1
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1answer
37 views

Diagonalizing a matrix M in an invariant subspace to find eigenvectors

Suppose we have a matrix M, and some subspace W which is invariant under M. Suppose W = span(a, b) where a and b are vectors. If we know what the action M on a and b results in (say some linear ...
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0answers
54 views

The value interpretation of eigenvectors.

My question is may be strange but I wanna lie it any way. The direction of an eigenvector is the most important as we normalize it. This view is right but what about the value of this eigenvector in ...

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