Tagged Questions
1
vote
3answers
51 views
How is my textbook finding this rotation?
I have this transformation $\mathbf x\mapsto A\mathbf x $ which is the composition of a rotation and a scaling. I need to give the angle $\varphi$ of the rotation and give the scale factor $r$. Here ...
0
votes
1answer
52 views
Two quick eigenvalues & complex numbers questions
A) For a vector $v\in\mathbb{C^n}$, is $Im(-v)=Im(\overline{v})$ ?
($Im(v)$denoting the imaginary part of the vector $v$)
My understanding: since every row of the vector is a complex number (say ...
0
votes
0answers
37 views
Determinant of a general circulant matrix
I'm dealing with a problem that is comparable to "How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?", yet slightly more difficult: I was asked to determine the ...
5
votes
1answer
84 views
What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?
Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.)
In ...
5
votes
2answers
71 views
Is $\mathrm{GL}_n(\mathbb C)$ divisible?
A group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Is the group $\mathrm{GL}_n(\mathbb C)$ divisible? Or more precisely, ...
3
votes
1answer
65 views
Does anyone know any resources for Quaternions for truly understanding them?
I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
0
votes
1answer
29 views
Need help interpreting an equation from an article (related to quaternions).
At this link, about half way down the page, there is an equation I don't understand
http://physicsforgames.blogspot.com/2010/02/quaternions-why.html
This is the equation.
$$VV† = -x^2I^2 - y^2J^2 - ...
4
votes
3answers
89 views
How do you construct the quaternion and the multiplication rules, like Hamilton did?
So, I understand complex number multiplication, and how it represents $2D$ rotations.
What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
3
votes
1answer
80 views
The multiplication of 2D vectors produces what?
I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication.
To avoid confusion with other types of multiplication, this is the basic form I ...
1
vote
2answers
66 views
Need help with matrix multiplication: $ (aI + bJ)(cI + dJ) $.
Consider the matrix
$$
A = \left[ \matrix{a & -b \\ b & a} \right],
$$
and write this as $ A = aI + bJ $, where
$$
I = \left[ \matrix{1 & 0 \\ 0 & 1} \right] \quad \text{and} \quad
J = ...
3
votes
2answers
41 views
What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?
In this article that talks about some history of hamilton
http://plus.maths.org/content/curious-quaternions
There is a snippet that says this:
Multiplication is very sneaky. You can only set up ...
3
votes
0answers
68 views
Prove (*) by induction on k.
Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form
$$\sum_{i=1}^m ...
-3
votes
4answers
93 views
Complete instead of Complex, Irregular instead of Imaginary
Will the terms complex and imaginary ever be replaced? At least within beginning classes?
I imagine it is more of a kind of hazing into the "mathemitician's club" to allow the terms to confuse ...
2
votes
2answers
62 views
Eigenvalues of a certain bordered identity matrix
Consider a complex $N-1 \times 1$ vector $b$ and a complex constant c. Let $I$ denote the $N-1 \times N-1$ identity matrix. Then what can we say about the eigenvalues of the matrix
\begin{align}
...
2
votes
1answer
119 views
Decompose a complex symmetric matrix to retain positive definitness
I have a complex symmetric matrix $A$, (i.e. non-Hermitian and obeying $A=A^T$), which is positive definite, in the sense that:
$$\Re({z^HAz}) > 0$$ for any $z$. I am able to verify this ...
0
votes
1answer
81 views
Finding a basis
Finding the basis for the kernel of:
\begin{pmatrix} a & b \\c & d\end{pmatrix}
$which$ $maps$ $to:$
\begin{pmatrix} a \\a\\3a + b \end{pmatrix}
It's all complex, but I'm not sure if ...
2
votes
2answers
193 views
Require brilliant resources to self teach.
I'm far from the level of mathematical knowledge every user on this website posseses, however I am very much determined to get there as my love for mathematics increases. These are the topics: ...
5
votes
2answers
155 views
is a one-by-one-matrix just a number (scalar)?
I was wondering. Clearly, we cannot multiply a (1x1)-matrix with a (4x3)-matrix; However, we can multiply a scalar with a matrix. This suggests a difference.
On the other hand, I was, for example, in ...
1
vote
2answers
82 views
Row reduction over any field?
EDIT: as stated in the first answer, my initial question was confused. Let me restate the question (I have to admit that it is now quite a different one):
Let's say we have a matrix $A$ with entries ...
0
votes
1answer
195 views
Complex number matrix calculation
How do I find a complex number $\lambda $ such that $\pmatrix{ 3&-2\\2&3}\vec v$ = $\lambda\vec v $ where $ \vec v $ is non-zero. Yes, this is a homework problem, I didn't learn complex number ...
6
votes
3answers
193 views
Help understanding $e^{it}=\cos t+i\sin t$ by way of matrices and vector fields
I was brushing up on my complex arithmetic in preparation for a class in ODE's this semester and I found myself looking at Exercise 2.7.5 in Introduction to Complex Analysis for Engineers by Michael ...
5
votes
1answer
211 views
Hermitian/positive definite matrices and their analogues in complex numbers
I've heard a couple of times some people say that in a way, Hermitian matrices are to matrices as real numbers are to complex numbers. I know two examples where this is sort of true:
Complex ...
13
votes
9answers
1k views
Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]
Possible Duplicate:
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
On Wikipedia, it says that:
Matrix ...
1
vote
0answers
48 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?
4
votes
2answers
354 views
Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant
If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$?
I remember seeing such a ...
3
votes
1answer
554 views
Are complex determinants for matrices possible and if so, how can they be interpreted?
I've been asked to compute the determinant of a 3x3 matrix with complex entries. I have done so using the normal expansion along a row or column method that I would use were the entries real. My ...
0
votes
0answers
73 views
Convert triangular real matrix to hermitian
We are developing some computer program which at some point uses a library (for which we do not have access to its source code) to solve the general eigenvalue problem; given two input real symmetric ...
1
vote
1answer
172 views
What is the relation between complex numbers and transformation matrices?
I read addition and multiplication with complex numbers can be represented as translation and rotation in a 2D plane.
I am using this to move around objects on the screen. I have an offset number, ...
5
votes
3answers
284 views
An application of Vandermonde determinant
Let $\lambda_1,\ldots,\lambda_n$ be complex numbers such that for each positive integer $k\geq 0$,
$$\sum_{i=1}^n \lambda_i^k=0.$$
Here I am supposed to show that $\lambda_i=0$ for each $i\in ...
0
votes
1answer
139 views
Finding a matrix that has complex Eigenvalues
I have an assignment where I need to create 2x2 matrices for each of the following Eigenvalue pairs.
...
12
votes
2answers
614 views
What's the name for the property of a function $f$ that means $f(f(x))=x$?
I can think of several examples of functions such that twice application of the function is equivalent to no application of it.
Additive inverse
Multiplicative inverse
Fourier transform
Complex ...
12
votes
4answers
384 views
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
I was reviewing some matrices and found this interesting
if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
0
votes
1answer
112 views
Generating a random Eisenstein integer matrix whose inverse has Eisenstein integer entries
Thanks to a question I previously asked, I realized that a Gaussian integer matrix should have a determinant of $\pm 1$ or $\pm i$ for it to have an Gaussian integer inverse. From that, I gather that ...
3
votes
0answers
112 views
On the distribution of unimodular matrices generated by the Hermite normal form
A problem I'm currently considering requires me to generate (pseudo-)random Gaussian integer matrices with Gaussian integer matrix inverses. By analogy with an algorithm I know for generating random ...
5
votes
2answers
181 views
Can a Gaussian integer matrix have an inverse with Gaussian integer entries?
Is there any way to characterize the set of complex matrices with Gaussian integer entries whose inverses also have Gaussian integer entries? I'm aware of the numerous examples of integer matrices ...
3
votes
3answers
3k 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
421 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 ...
6
votes
1answer
860 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 ...
