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 ...
0
votes
1answer
36 views

Extension to the complex numbers for ex. 12 in ch. 6 of Axler's “Linear Algebra Done Right”

I'm wondering how the answer to Sheldon Axler's exercise 12 of chapter 6 "Linear Algebra Done Right" changes when the underlying field is extended from the reals to the complex numbers. The exercise ...
0
votes
3answers
55 views

Complex Numbers and Linear Algebra

Explain why there does not exist a $\lambda $ in the Complex Field such that $$\lambda \left(2-3i, 5+4i, -6+7i \right) = \left(12-5i, 7+22i, -32-9i \right)$$ Can someone help me figure out how to go ...
0
votes
1answer
65 views

Complex numbers: How to solve the “contradiction”? [duplicate]

$$-1 = i\cdot i = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$$ $$-1 = 1$$ Obviously, something is wrong here, but I can't put my finger on it. How to solve this "contradiction"?
0
votes
0answers
21 views

Is alpha and endomorphism of C considered as avector space over R? Is it an endomorphism of C considered as avector space over itself?

Let alpha:C$\to$C be the function defined by alpha:a+bi$\to$ -b+ai.(1) Is alpha and endomorphism of C considered as a vector space over R?(2) Is it an endomorphism of C considered as a vector space ...
1
vote
3answers
32 views

Query regarding Linear Transformation…

As we always read in Complex Analysis, Linear Transformation (L.T.) is a combination of Translation, Rotation and Magnification i.e. $T(z)=az+b$ is a L.T. in complex. However, It doesn't satisfy the ...
-1
votes
2answers
42 views

Which is the Hermitian inner product, in terms of conjugate and transpose?

Page 29 of Source 1: Denote the complex conjugate by * : $\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$ Page 1 of Source 2: $\mathbf{u \cdot v} = ...
-2
votes
2answers
31 views

Find the roots of the simple equation?

x^{2}= 0 What are the roots? are they in complex plane, but how? Answer seems trivial in real numbers ain't it? Does this evolve a new system like it was with iota?
0
votes
2answers
36 views

Logs of a complex number

Write a solution in Cartesian for of What should come next?
1
vote
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 ...
2
votes
3answers
29 views

Prove that $|a|+|b|\le \sqrt{2}|z|$

I was solving maths and got struk on this question.might you help me with this one. If z=a+ib Then, prove that $|a|+|b|\le \sqrt{2}|z|$ I don't know how to start it. Help me.
0
votes
1answer
29 views

Linear Algebra,Conjugate Transpose

Let $ M_n(\mathbb C) $ be the space of all $ n\times n $ matrices with complex entries. Prove that function $ \langle, \rangle : M_n(\mathbb C) \times M_n(\mathbb C) \to \mathbb C $ defined by $ ...
3
votes
1answer
63 views

Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
1
vote
1answer
28 views

solve in terms of complex numbers

I need the full solution (with steps) of $K^4=-4$. First, I tried to solve in termes of $K^2$ and I tried to include in my answer the j term of complex numbers. Thanks
0
votes
1answer
33 views

Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?

I wonder about the second step of the proof shown below (d) in the picture attached. Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?
0
votes
1answer
20 views

Is $\|x_1\|^2 + 2\|x_2\|^2 > - 2\Re(ix_1\overline{x_2})$ for complex numbers $x_1,x_2$

This is the last piece I need for a proof for a homework problem. Could someone explain whether or not this inequality must hold?
0
votes
1answer
59 views

How to deal with $\bar{x}$ when solving complex-variable linear equation(s) of x?

The theory of linear algebra can be directly applied to linear equation(s) of complex variables with the form \begin{equation} \sum_i a_i x_i=c\ldots\ldots(1) \end{equation} with $a_i,c\in ...
0
votes
1answer
56 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 ...
0
votes
1answer
28 views

Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
0
votes
1answer
35 views

Locus in Complex plane

Could someone help me out with this one Show that the locus of w as z varies with |z| = 1, where w is given by $$w^2=\frac {1-z}{1+z}$$ is a pair of straight lines.
1
vote
4answers
46 views

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$?

If $a \in \mathbb{C}$, is $|a|^2=\bar{a}a=a\bar{a} \in \mathbb{R}$? Meaning, if I have a complex number and I multiply it by its complex conjugate, would that always return a number in $\mathbb{R}$? ...
2
votes
3answers
191 views

solving complex numbers with powers algebraically

Find algebraically the value of :$\left(2^{0.5} + 6^{0.5} - \left( 2^{0.5} - 6^{0.5} \right)i \right)^4$ Below are my works I try to simplify inside. but i found that i can't add $2^{0.5}$ and ...
0
votes
1answer
24 views

Finding solution in polar form, raised to a power

The problem asks, find an equation equal to: z^3 = ( 1 + sqrt(3)*i ) where i is the square root of negative one. I tried approaching this problem first by ...
2
votes
2answers
122 views

Solve the equations $z^2 + (2 - 2i)z + 2i = 0 $ by completing the square

I tried solving this thing by completing the square and I always end up with something like this $(z^2 + (2 - 2i)z - 2i) + 2i + 2i = 0 $ and it doesn't seem like to me that you can factor the part in ...
1
vote
0answers
17 views

Alternative coordinates for the complex plane $\mathrm{Re}[e^{-is}z]=a$, $\mathrm{Re}[e^{-it}z]=b $

I am defining coordintes on $\mathbb{C}$ using a "generalized" real and imaginary part. Here $a,b \in \mathbb{R}$. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ...
0
votes
2answers
38 views

Proving an inequality with the Schwarz inequality

Given a vector space with a Hermitian dot product defined, prove the following inequality using the Schwarz inequality. Let $f$ be a complex value function that is continuous within $0 \le x \le 1$, ...
0
votes
1answer
18 views

Linear Algebra Complex values

The cube roots of $-3+2i$ are $x_1 = (1.0106+1.1532i),\; x_2 = (0.4934-1.4519i),\text{ and }x_3 = (-1.5040+bi)$ What is $b$? So $$-3+2i = (x_1)(x_2)(x_3) = -3.268 + 2.172bi + 1.351i + 0.898b$$ ...
2
votes
3answers
212 views

Linear Algebra Complex Numbers

The solutions to the equation $z^2-2z+2=0$ are $(a+i)$ and $(b-i)$ where $a$ and $b$ are integers. What is $a+b$? I simplified and got $(z+1)(z+1) = -1$ and now I'm not sure where to go from there. ...
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
vote
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 ...
1
vote
1answer
66 views

Must unitary matrices satisfying this property commute?

If A and B are unitary matrices such that A, B, and AB are all conjugate to diag(1,1,-1,-1), must AB=BA? Why or why not?
4
votes
2answers
87 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
487 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 ...
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
vote
1answer
41 views

Calculate the inverse of a complex matrix

I am trying to calculate the inverse of a given matrix but somewhere I have an error in my calculation that I cannot find $$\begin{array}{ccc} && \left( \begin{array}{ccc|ccc} 1-i & 2 ...
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 $A$.
1
vote
2answers
160 views

Factor the polynomial $x^3 − 27$ using De Moivre's theorem (Please explain solution)

I was reading the book A First Course in Linear Algebra by Ken Kuttler (link to nearly identical page http://librarum.org/book/312/11) and I did not understand this part: Q: Factor the polynomial ...
4
votes
4answers
93 views

Prove that for vectors $v_1,…,v_n$ in $\mathbb C^n$, $\{v_1,…,v_n\}$ is a basis for $\mathbb C^n$ iff its conjugate is a basis for $\mathbb C^n$

Prove that for vectors $v_1,...,v_n$ in $\mathbb C^n$, $\{v_1,...,v_n\}$ is a basis for $\mathbb C^n$ if and only if $\{\bar v_1,..., \bar v_n\}$ is a basis for $\mathbb C^n$. I know intuitively that ...
1
vote
1answer
55 views

Determining the polar form for all n-th roots of unity.

By definition $ z \in\mathbb{C}$ is a n-th root of unity iff $z^n = 1$. My assignment is to (iv) List all n-th roots of unity in their polar form. You may use that there are only $n$ Elements with ...
1
vote
1answer
24 views

Parametrized complex equation

What are the values that satisfy the following equation: $$\frac{z+i}{\bar{z}-2}=-2i$$ I have tried substituting $z$ with $a+bi$, and multiplying the left hand side by $\frac{z-2}{z-2}$ but the ...
1
vote
0answers
37 views

Show a polynomial is reducible to linear terms - check my answer

I have an exam tomorrow in linear algebra, and I want to make sure I answered this question correctly. Let $p \in \mathbb R[x], z \in \mathbb{C}$. We are given if $Im(z)>0$ then $p(z)\neq0$ Show ...
1
vote
4answers
106 views

Complex numbers problem

I have to solve where n is equal to n=80996.
1
vote
2answers
44 views

Make the vector $[1,1]$ turn of an angle - $\pi/4$ , with complex numbers

We have $[1,1]$ and $\theta = -\pi/4$ here is my attempt: $(\cos(-\pi/4) + i \sin(-\pi/4)) * (x+iy)$ = $(\sqrt{2}/2 - i \sqrt{2}/2) (1+i)$ = $\sqrt{2}/2 - i^2\sqrt{2}/2 $ = $[\sqrt{2}/2 + ...
3
votes
1answer
49 views

Triangle inequality- complex

I am trying to prove the triangle inequality purely algebraically. Let $z=x+iy$, $w=u+iv$. Then, $|z+w|^2$=$|(x+u)+i(y+v)|^2$=$(x+u)^2+(y+v)^2$=$x^2+2xu+u^2+y^2+2yv+v^2$ I tried the other way: ...
2
votes
1answer
225 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
0answers
26 views

Visualising $aw_1+bw_2=c \text{ where } a,b,c\in\mathbb R \quad w_1,w_2\in\mathbb C$

$aw_1+bw_2=c \text{ where } a,b,c,w_1,w_2\in\mathbb R, $ Is wel known and well studied by pupils very early on However I do not recall having seen $aw_1+bw_2=c \text{ where } a,b,c\in\mathbb R \quad ...
1
vote
1answer
45 views

Quantum Fourier Transform and roots of unity.

I need to find $QFT_{6}$ for the state quantum state $\frac{1}{\sqrt2}(|0\rangle + |3\rangle)$. I received a very sufficient answer recently on simplifying nth roots of unity, but I am having a lot of ...
1
vote
2answers
170 views

If C is the set of complex numbers, how do you show that C is a vector space with given operations?

I am presented with the question: Let $\mathbb{C}$ be the set of complex numbers. Define addition on $\mathbb{C}$ by: $(a+bi)+(c+di)=(a+c)+(b+d)i$. Define scalar multiplication by: $\alpha\cdot ...