0
votes
1answer
15 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
17 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
110 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
65 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
vote
1answer
42 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
147 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
vote
2answers
46 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
vote
2answers
70 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
42 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
176 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
502 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
vote
2answers
4k 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
111 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
76 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
70 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 ...
1
vote
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?
0
votes
1answer
324 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
382 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 ...
5
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
vote
2answers
709 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 ...
8
votes
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 ...