1
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1answer
72 views

Eigenvalues and Eigenvectors for matrix. Complex Eigenvalues

How can I find out the eigenvectors for this matrix: $$A= \begin{pmatrix} -3 &0&0\\ 0&3&-2\\ 0&1&1 \end{pmatrix} $$ I found the eigenvalues: $\lambda_{1}=-3$, ...
0
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1answer
69 views

Convergence rate of PageRank, the problem when the second eigenvalue is complex

As far as I know the Google matrix used to calculate the PageRank is not symetric, that means that some eigenvalues can be complex, furthermore, we know that the second eigenvalue is equal to the ...
1
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0answers
40 views

Eigenvalues of complex matrix

I'm taking linear algebra II this semester and the course assumes that students have already covered complex numbers. Unfortunately I take my first analysis course, in which complex numbers are ...
0
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1answer
21 views

Complex Eigenvalue Periodic

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue: $$\lambda = \alpha + i\beta$$ I know that $\alpha < 0$ ...
0
votes
0answers
21 views

Relationship between two sets of eigenvectors.

Given the eigenvalue problems \begin{align} \mathbf{A} \vec{x} &= \lambda \vec{x}, \\ \mathbf{A}\left[\mathbf{S}^{-1}\mathbf{P}\vec{x}^*\right] &= ...
4
votes
2answers
151 views

Can an idempotent matrix have complex eigenvalues?

Let $P\in\mathbb{R}^{n\times n}$ be a nontrivial idempotent matrix: $P^2=P$, $P\neq 0$, $P\neq I,$ where $I$ is the $n\times n$ identity matrix. What are the eigenvalues of $P$? Solution: Let $x$ be ...
2
votes
1answer
69 views

$A+A^T=I$, $\lambda$ is an eigenvalue of $A$, show that $\lambda=\frac{1}{2}+\alpha i$

I tried to solve it but I got $\lambda =\frac{1}{2}$ without the complex part, I'd like to know where my logic is flawed. Assume $v$ is the eigenvector associated with lambda, then: $(A+A^T)v=Iv$ ...
1
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1answer
47 views

Calculating the eigenvalues of a given matrix, please check my results

Given the matrix $$A = \left(\begin{array}{ccc} 1&-0.85&0\\ 1.7&-1&0\\ 0 & 0.85 & 4 \end{array}\right)\in\mathbb{C}^{3\times3}$$ I am now looking for the ...
5
votes
2answers
333 views

Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
1
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2answers
51 views

Complex Matrix Limit

If $A$ is an $n \times n$ complex matrix, show that if $\lim_{k\rightarrow\infty}||A^kv||=0$ for every vector $v \in \Bbb C^n$, then $|\lambda|\leq1$ for every eigenvalue $\lambda$ of $A$.
1
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2answers
73 views

What's the formula matrix with complex numbers?

Given$$\mathbf{M}= \begin{pmatrix} 7 & 5 \\ -5 & 7 \\ \end{pmatrix} $$, what's the formula matrix for $\mathbf{M}^n$? The eigenvalues and eigenvectors are ...
0
votes
1answer
46 views

Are complex eigenvalues special?

I've noticed that complex eigenthings are treated as a whole separate topic to real eigenthings (I say things to mean values and vectors). I see no reason for this distinction, yes it makes the ...
0
votes
0answers
216 views

Determining rotation axis for matrix with complex eigenvectors.

I'm using Zhang's method to determine the 3D camera parameters from a set of images. When calculating extrinsic parameters for the third image, I get the following matrix. $$ \begin{vmatrix} ...
0
votes
1answer
751 views

Finding diagonal and unitary matrices

Let $A=\begin{pmatrix} 1 & 1+i\\ 1-i & 2 \end{pmatrix}$ I'm trying to find a diagonal matrix $D$ and a unitary matrix $U$ so that $U^\star AU=D$. (We define $U^*=\overline{U}^t$ ). I ...
1
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2answers
5k views

How to determine if a matrix is positive/negative definite, having complex eigenvalues?

I am trying to deal with an issue: I am trying to determine the nature of some points, that's why I need to check in Matlab if a matrix with complex elements is positive or negative definite. After ...
1
vote
1answer
117 views

What are the Complex (Non-Real) Eigenvectors of $3\times 3$ Rotation Matrices?

A $3\times 3$ rotation matrix $R$ that rotates $\mathbb{R}^3$ around the unit vector $v\in\mathbb{R}^3$ by angle $\theta$ (as defined by Rodrigues' rotation formula) satisfies the ...
2
votes
2answers
81 views

Eigenvalues of a certain bordered identity matrix

Consider a complex $N-1 \times 1$ vector $b$ and a complex constant c. Let $I$ denote the $N-1 \times N-1$ identity matrix. Then what can we say about the eigenvalues of the matrix \begin{align} ...
2
votes
2answers
73 views

eigenvalues for a complex matrix

How do you find the eigenvalues (hence the eigenvectors too) of a matrix with complex bits like this: $$\hat{H}=\epsilon \begin{vmatrix} 0&i&0 \\\\ -i&0&0 \\\\ 0&0&-i ...
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0answers
57 views

complex eigenvalues of matrix sum

I have a set of (about 100) general real square matrices. Is it possible to determine whether none of their linear combinations has complex eigenvalues?
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1answer
365 views

Finding a matrix that has complex Eigenvalues

I have an assignment where I need to create 2x2 matrices for each of the following Eigenvalue pairs. ...
0
votes
1answer
403 views

Complex conjugates and complex eigenvalues

Suppose the matrix A with real entries has complex eigenvalues $\lambda = \alpha + i\beta\,$ and $\overline{\lambda} = \alpha - i\beta\,$. Suppose that $Y_0 = (x_1 + iy_1, x_2 + iy_2)$ is an ...
6
votes
3answers
5k views

Can a real symmetric matrix have complex eigenvectors?

A Hermitian matrix always has real eigenvalues and real or complex orthogonal eigenvectors. A real symmetric matrix is a special case of Hermitian matrices, so it too has orthogonal eigenvectors and ...
1
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2answers
747 views

Raising a square matrix to the k'th power: From real through complex to real again - how does the last step work?

I am reading Applied linear algebra: the decoupling principle by Lorenzo Adlai Sadun (btw very recommendable!) On page 69 it gives an example where a real, square matrix $A=[(a,-b),(b,a)]$ is raised ...
10
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1answer
1k views

Intuition for complex eigenvalues

The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same. My question What is the ...