# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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"?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?
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### 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?
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The theory of linear algebra can be directly applied to linear equation(s) of complex variables with the form $$\sum_i a_i x_i=c\ldots\ldots(1)$$ with $a_i,c\in ... 1answer 56 views ### Matrix representation of complex numbers in exponential form Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ... 1answer 28 views ### Inner product over the$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, ... 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. 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$\mathbb{R}$? ... 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 ... 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 ... 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 $and it doesn't seem like to me that you can factor the part in ... 0answers 17 views ### Alternative coordinates for the complex plane$\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}$. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ... 2answers 38 views ### Proving an inequality with the Schwarz inequality Given a vector space with a Hermitian dot product defined, prove the following inequality using the Schwarz inequality. Let$f$be a complex value function that is continuous within$0 \le x \le 1$, ... 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 $$-3+2i = (x_1)(x_2)(x_3) = -3.268 + 2.172bi + 1.351i + 0.898b$$ ... 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. ... 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:$(A+A^T)v=Iv$... 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 ... 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? 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 ... 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 ... 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 ... 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 ... 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 $$\begin{array}{ccc} && \left( \begin{array}{ccc|ccc} 1-i & 2 ... 2answers 51 views ### Complex Matrix Limit If A is an n \times n complex matrix, show that if \lim_{k\rightarrow\infty}||A^kv||=0 for every vector v \in \Bbb C^n, then |\lambda|\leq1 for every eigenvalue \lambda of A. 2answers 160 views ### Factor the polynomial x^3 − 27 using De Moivre's theorem (Please explain solution) I was reading the book A First Course in Linear Algebra by Ken Kuttler (link to nearly identical page http://librarum.org/book/312/11) and I did not understand this part: Q: Factor the polynomial ... 4answers 93 views ### Prove that for vectors v_1,…,v_n in \mathbb C^n, \{v_1,…,v_n\} is a basis for \mathbb C^n iff its conjugate is a basis for \mathbb C^n Prove that for vectors v_1,...,v_n in \mathbb C^n, \{v_1,...,v_n\} is a basis for \mathbb C^n if and only if \{\bar v_1,..., \bar v_n\} is a basis for \mathbb C^n. I know intuitively that ... 1answer 55 views ### Determining the polar form for all n-th roots of unity. By definition z \in\mathbb{C} is a n-th root of unity iff z^n = 1. My assignment is to (iv) List all n-th roots of unity in their polar form. You may use that there are only n Elements with ... 1answer 24 views ### Parametrized complex equation What are the values that satisfy the following equation:$$\frac{z+i}{\bar{z}-2}=-2i$$I have tried substituting z with a+bi, and multiplying the left hand side by \frac{z-2}{z-2} but the ... 0answers 37 views ### Show a polynomial is reducible to linear terms - check my answer I have an exam tomorrow in linear algebra, and I want to make sure I answered this question correctly. Let p \in \mathbb R[x], z \in \mathbb{C}. We are given if Im(z)>0 then p(z)\neq0 Show ... 4answers 106 views ### Complex numbers problem I have to solve where n is equal to n=80996. 2answers 44 views ### Make the vector [1,1] turn of an angle - \pi/4 , with complex numbers We have [1,1] and \theta = -\pi/4 here is my attempt: (\cos(-\pi/4) + i \sin(-\pi/4)) * (x+iy) = (\sqrt{2}/2 - i \sqrt{2}/2) (1+i) = \sqrt{2}/2 - i^2\sqrt{2}/2 = [\sqrt{2}/2 + ... 1answer 49 views ### Triangle inequality- complex I am trying to prove the triangle inequality purely algebraically. Let z=x+iy, w=u+iv. Then, |z+w|^2=|(x+u)+i(y+v)|^2=(x+u)^2+(y+v)^2=x^2+2xu+u^2+y^2+2yv+v^2 I tried the other way: ... 1answer 225 views ### Inverse of a real matrix plus identity times i How would you proof that given a real square matrix A then the inverse of the matrix ( A + i I ) exists? 1answer 49 views ### Does i^T=-i or i^T = i ?(T is transpose) Assume A is a skew-symmetric matrix(its eigenvalues is zero or purely imaginary). If x is its eigenvector, we have$$x^TAx=\lambda|x|^2$$, take transpose on both sides. we have ... 0answers 26 views ### Visualising$aw_1+bw_2=c \text{ where } a,b,c\in\mathbb R \quad w_1,w_2\in\mathbb Caw_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 ...
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 ...
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 ...