0
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0answers
11 views

Complex Matrix Orientation

I recently learned about the fact that a linear mapping of a real vector space is orientation-preserving if the determinant of the matrix is positive. Now I was wondering if there exists a similar ...
1
vote
1answer
24 views

Generating the special linear group of 2 by 2 matrices over the integers.

Our Number Theory professor claimed that the special linear group $\text{SL}_2(\mathbb{Z})$ is generated by just two matrices: $$ M_1=\begin{pmatrix} 0& -1\\ 1& 0 \\\end{pmatrix} $$ $$ ...
3
votes
0answers
60 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
2
votes
3answers
71 views

Real matrices with non-real eigenvalues

I know this covers a lot, so perhaps someone could redirect me to a helpful website. for a) I have no idea where to start on the proof, as I don't understand why this is true. for b) I also have ...
2
votes
2answers
59 views

2x2 inverse of a complex matrix with complex determinant

Firstly, my question may be related to a similar question here: Are complex determinants for matrices possible and if so, how can they be interpreted? I am using: $$ \left(\begin{array}{cc} a&b\\ ...
0
votes
1answer
19 views

Complex function and Jacobian matrix

Given some complex-differentiable function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined $f(x,y)=u(x,y)+iv(x,y)$, we know the Cauchy-Riemann equations hold, so: $$\dfrac{\partial u}{\partial ...
0
votes
1answer
27 views

2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through: My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts ...
1
vote
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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0answers
37 views

Comparison of two matrix multiplication operations.

I am comparing the below operations: $$ A=\begin{bmatrix} a & 0 & f & g \\ 0 & b & 0 & 0 \\ f & 0 & c & h \\ g & 0 & ...
1
vote
1answer
42 views

Matrices and Complex Numbers [duplicate]

Given this set: $$ S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\} $$ Part I: Why is this set equivalent to the set of all complex numbers a+bi (when both are ...
0
votes
1answer
71 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 ...
0
votes
1answer
57 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
1
vote
2answers
69 views

What does it mean to multiply a real matrix by a complex scalar?

In this answer http://math.stackexchange.com/a/219508/27609 it is noted, that multiplying a matrix $A$ by a scalar $s$ is the same as multiplying a matrix $A$ by a diagonal matrix ${\rm ...
2
votes
2answers
82 views

Simplfy a complex matrix into a real one

I encounter systems of linear complex equations (At most 3 equations) in my circuit analysis course. The calculator I am using is Casio fx-991ES and it only accepts real elements when in matrix or ...
0
votes
2answers
46 views

Proving a Simple equation

I have a not so smart question; but I just cannot figure it out ! Suppose that I have a real $2 \times 2 $ matrix $(a_{ij})$ of non-zero determinant, and let $z \in \mathbb{C} $ be such that $ ...
0
votes
0answers
11 views

Quatornianic Matrix are Complex, Proposition

In my matrix group textbook, I was looking at a proposition that said For all $\lambda \in \mathbb R$ and $A, B \in M_n(\mathbb H)$, $\Psi_n(\lambda \bullet A) = \lambda \bullet \Psi_n(A)$ Is this ...
2
votes
1answer
70 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: $(A+A^T)v=Iv$ ...
2
votes
3answers
48 views

Quaternion identity proof

If $q \in \mathbb{H}$ satisfies $qi = iq$, prove that $q \in \mathbb{C}$ This seems kinda of intuitive since quaternions extend the complex numbers. I am thinking that $q=i$ because i know that $ij = ...
4
votes
2answers
88 views

$T=-T^{*}$, show that $T+\alpha I$ is invertible.

Please don't answer the question. Just tell me if I am in the right direction. I should be able to solve this. We are given $T=-T^{*}$, show that $T+\alpha I$ is invertibe for all real alphas that ...
1
vote
3answers
133 views

short question regarding convention - symmetric matrices and transpose

I have a short question because wikipedia is extremly vague on this subject. Suppose I have the matrix $A=\begin{pmatrix} i & 1 \\ 1 & -i\end{pmatrix}$. Is it symmetric? I mean, in the ...
0
votes
1answer
529 views

How do I find transformation matrix with respect to standard basis?

I know that in order to find transformation matrix with respect to a basis, I need to apply the transformation to said basis and the result is the column of the transformation matrix. But what ...
1
vote
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 $A$.
0
votes
1answer
300 views

complex numbers and 2x2 matrices

Is it correct that set ${\mathbb C}$ is isomorphic to the set of following 2x2 matrices: $$\left( \begin{array}{cc} a &-b\\ b &a \end{array}\right) $$ $a \in {\mathbb R}$ and $b \in {\mathbb ...
3
votes
1answer
51 views

Are there matrices with with non-real elements?

I know that the definition of a matrix is a rectangular arrangement of elements, which are real numbers. But does there exist such a thing as a rectangular arrangement of complex numbers? How are ...
1
vote
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
44 views

Representing complex numbers as matrices, show that $A(z)+A(z')=A(z+z')$

I am doing a task where in which I am representing complex numbers as matrices, so $z=x+iy \in \Bbb C$ is represented by: $A(z)=\begin{bmatrix} x & -y \\ y & x \end{bmatrix}$ Now I have to ...
2
votes
1answer
236 views

Inverse of a real matrix plus identity times i

How would you proof that given a real square matrix $A$ then the inverse of the matrix ( $A + i I $) exists?
2
votes
1answer
49 views

Does $i^T=-i$ or $i^T = i$ ?(T is transpose)

Assume $A$ is a skew-symmetric matrix(its eigenvalues is zero or purely imaginary). If $x$ is its eigenvector, we have $$x^TAx=\lambda|x|^2$$, take transpose on both sides. we have ...
0
votes
1answer
71 views

Eigenvectors of a symmetrical matrix

Accepting the fact that, as all others normal matrices, a hermitian matrix (with N lines) has a set of N orthonormal complex eigenvectors (with real eigenvalues, degenerate or not), how could I prove ...
5
votes
0answers
82 views

Complex Numbers vs. Matrix

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by ...
0
votes
1answer
213 views

rank of complex conjugate transpose matrix property proof

I have a question about complex conjugate matrix. Prove that for any rectangular matrix $A$, rank $A$=rank $A^*$ where $A^*$ is complex conjugate transpose of A.
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vote
0answers
47 views

Instead of iteration method, can we solve this matrix computation?

I have a simple problem with matrix compuation. Matices what I have are $A_f$ matrix with $(M \times N)$, $B_f$ matrix(or vector) with $(N \times 1)$. I just want to calculate $|A_fB_f|^2$ for $F$ ...
0
votes
2answers
91 views

Complex exponents and matrices

If the matrix $A$ is defined as: $$A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & ...
1
vote
3answers
129 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 ...
2
votes
1answer
102 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
278 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
111 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
83 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, ...
5
votes
2answers
472 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
35 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
311 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
183 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
121 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
59 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
96 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 ...
-2
votes
4answers
102 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
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
1answer
268 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
85 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 ...
3
votes
2answers
283 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: ...