# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $A+A^T=I$, $\lambda$ 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: $(A+A^T)v=Iv$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 $F$ ...
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### 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 ...
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### 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 ...
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Consider the matrix $$A = \left[ \matrix{a & -b \\ b & a} \right],$$ and write this as $A = aI + bJ$, where $$I = \left[ \matrix{1 & 0 \\ 0 & 1} \right] \quad \text{and} \quad J = ... 2answers 59 views ### What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension? In this article that talks about some history of hamilton http://plus.maths.org/content/curious-quaternions There is a snippet that says this: Multiplication is very sneaky. You can only set up ... 0answers 96 views ### Prove (*) by induction on k. Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form$$\sum_{i=1}^m ...
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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 ...
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### 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} ...
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### 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 ...
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 ...