Matrices Definition and 1000 Threads

Hi, I am studying about circulant matrices, and I have seen that one of the properties of such matrices is the eigenvalues which some combinations of roots of unity. I am trying to understand why it is like that. In all the places I have searched they just show that it is true, but I would like...

This is something I seek a proof of. Theorem: Let ## \mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}## be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous. My attempt. Continuity must be judged in...

Homework Statement Homework Equations using nodal analysis The Attempt at a Solution https://imgur.com/a/UNEDH the excel sheet is the matrix i set up then used cramer's rule. I think i got the method down but i just can't find the error so hopefully i overlooked something. The equation i...

If there is matrix that is formed by blocks of 2 x 2 matrices, what will be the relation between the eigen values and vectors of that matrix and the eigen values and vectors of the sub-matrices?

Homework Statement Homework Equations A.A^-1=Identity matrix Trace of a matrix [tr(A)]is the sum of elements on it's main diagonal. The Attempt at a Solution In the given equation,post-multiplying A^-1 (A inverse) on both sides gives A^4=16. Since the array contains only one element (say...

Can anyone give me geometric and intuitive insight on Rotation matrices which has two sets of coordinates after Transformation?

Here it is a simple problem which is giving me an headache,Recall from class that in order to build an invariant out of spinors we had to introduce a somewhat unexpected form for the dual spinor, i.e. ߰ψ = ψ†⋅γ0 Then showing that ߰ is invariant depends on the result that (ei/4⋅σμν⋅ωμν)† ⋅γ0 =...

Homework Statement We can treat the following coupled system of differential equations as an eigenvalue problem: ## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ## ## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ## ## \frac{dy_3}{dt} = f_3 - 4y_3 ## where f1, f2 and f3 is a set of time-dependent sources, and...

In Lancaster & Burnell book, "QFT for the gifted amateur", chapter 48, it is explained that, with a special set of ##\gamma## matrices, the Majorana ones, the Dirac equation may describe a fermion which is its own antiparticle. Then, a Majorana Lagrangian is considered...

Homework Statement About an endomorphism ##A## over ##\mathbb{C^{11}}## the next things are know. $$dim\, ker\,A^{3}=10,\quad dim\, kerA^{2}=7$$ Find the a) Jordan canonical form of ##A## b) characteristic polynomial c) minimal polynomial d) ##dim\,kerA## When: case 1: we know that ##A## is...

Homework Statement Let ##A = \begin{bmatrix} a&b\\c&d \end{bmatrix}## such that ##a+b+c+d = 0##. Suppose A is a row reduced. Prove that there are exactly three such matrices. Homework EquationsThe Attempt at a Solution 1) ##\begin{bmatrix} 0&0\\0&0 \end{bmatrix}## 2) ##\begin{bmatrix}...

Hi there, This is a question about numerical analysis used particularly in the computational condensed matter or anywhere where one needs to DIAGONALIZE GIGANTIC DENSE HERMITIAN MATRICES. In order to diagonalize dense Hermitian matrices size of 25k-by-25k and more (e.g. 1e6-by-1e6) it is not...

For finding the inverse of a matrix A, we convert the expression A = I A (where I is identity matrix), such that we get I = B A ( here B is inverse of matrix A) by employing elementary row or column operations. But why do these operations work? Why does changing elements of a complete row by...

So I have been studying the case of spin 1/2 and I have understood how the formulations work through to find the spin matrices. However I do not get an intuitive understanding of what they mean and why they are formulated the way they are. I follow Griffith's book and in it as he begins to solve...

I have a word problem that I am struggling with. I have been using matrices in this chapter, but I don't understand how it applies or where to start in order to solve this equation. Here is the word problem: One hundred liters of a 50% solution is obtained by mixing a 60% solution with a 20%...

Hello! I am not really sure I understand the idea of tensors and the difference between them and normal matrices, for example (for rank 2 tensors). Can someone explain this to me, or give me a good resource for this? I don't want a complete introduction to GR math, I just want to understand the...

Let's say A is a singular matrix. Will the transpose of this matrix be always singular? If so why?

Hi. If I have an operator in matrix form eg. < i | x | j > then the matrix of the operator x2 is given by the square of the former matrix. This seems like common sense but how would i prove this using Dirac notation ? Thanks

Hi, I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra). 1. Homework Statement Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##...

Homework Statement The vectors ##a_1, a_2, a_3, b_1, b_2, b_3## are given below $$\ a_1 = (3~ 2~ 1 ~0) ~~a_2 = (1~ 1~ 0~ 0) ~~ a_3 = (0~ 0~ 1~ 0)~~ b_1 = (3~ 2~ 0~ 2)~~ b_2 = (2 ~2~ 0~ 1)~~ b_3 = (1~ 1~ 0~ 1) $$ The subspace of ## \mathbb R^4 ## spanned by ##a_1, a_2, a_3## is denoted by...

Hello, I would just like some help clearing up some pretty basic things about hermitian operators and matricies. I am aware that operators can be represented by matricies. And I think I am right in saying that depending on the basis used the matrices will look different, but all our valid...

So I know that in general, for the ring of ##n \times n## matrices, if ##AB = 0##, then it is not necessarily true that ##A=0## or ##B=0##. However, in other rings, for example the integers ##\mathbb{Z}##, I know that this statement is true. So what property is the ring of matrices lacking such...

Homework Statement Let ##A## and ##B## be ##n \times n## matrices 1) Suppose ##A^2 = 0##. Prove that ##A## is not invertible. 2) Suppose ##AB=0##. Could ##A## be invertible. 3) If ##AB## is invertible, then ##A## and ##B## are invertible Homework EquationsThe Attempt at a Solution 1) Suppose...

Homework Statement Consider the subspace $$W:=\Bigl \{ \begin{bmatrix} a & b \\ b & a \end{bmatrix} : a,b \in \mathbb{R}\Bigr \}$$ of $$\mathbb{M}^2(\mathbb{R}). $$ I have a few questions about how this can be decomposed. 1) Is there a subspace $$V$$ of...

Homework Statement [/B] I know the pauli matrices in terms of the z-basis, but can't find them in terms of the other bases. I would like to know what they are. Homework Equations The book says they are cyclic, via the relations XY=iZ, but this doesn't seem to apply when I use this to find the...

Homework Statement Question is uploaded I have completed till part iii and obtained correct answers i. 2 ii. Basis for R:- { ( 2 3 -1 ) , (1 4 2 ) } Cartesian equation; 2x-y+z=0 iii. Basis for Null:- { ( -3 2 0 1 ) , (2 -3 1 0 ) } 2. The attempt at a solution I have problem in last part. I...

1. Problem statement : suppose we have a Hermitian 3 x 3 Matrix A and X is any non-zero column vector. If X(dagger) A X > 0 then it implies that determinant (A) > 0. I tried to prove this statement and my attempt is attached as an image. Please can anyone guide me in a step by step way to...

Homework Statement Let ##A## and ##B## be different ##n \times n## with real entries. If ##A^3 = B^3## and ##A^2 B = B^2 A##, can ##A^2 + B^2## be invertible? Homework EquationsThe Attempt at a Solution So, first of all I am just trying to interpret the question correctly. Does "can ##A^2 +...

Homework Statement Let A,B,C,D be commuting n-square matrices. Consider the 2n-square block matrix ##M= \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}##. Prove that ##\left | M \right |=\left | A \right |\left | D \right |-\left | B \right |\left | C \right |##. Show that the result may not be...

Homework Statement Suppose B is a real nonsingular matrix. Show that: (a) BtB is symmetric and (b) BtB is positive definite 2. Homework Equations N/A The Attempt at a Solution I have managed to prove (a) by showing that elements that are symmetric on the diagonal are equal. However I have no...

Is it possible to find matrices that commute but eigenvectors of one matrix are not the eigenvectors of the other one. Could you give me example for it?

I have the following system of equations: ##2t-4s=-2;~-t+2s=-1;~3t-5s=3##. With them, I form the matrix \begin{bmatrix} 2 & -4 & -2 \\ -1 & 2 & -1 \\ 3 & -5 & 3 \end{bmatrix} Which turns out to be row equivalent to \begin{bmatrix} 1 & 0 & 11 \\ 0 & 1 & 6 \\ 0 & 0 & 0 \end{bmatrix} so...

Homework Statement Consider the following matrix where * indicates an arbitrary number and a ■ indicates a non zero number. http://prntscr.com/e4xqkx [■ * * * * | *] [0 ■ * * 0 |* ] [0 0 ■ * * | *] [0 0 0 0 ■ | *] (Sorry for poorly formatted matrix. The link above contains a screenshot...

Homework Statement Let ##V\subset \mathbb{R}^3## be the subspace generated by ##\{(1,1,0),(0,2,0)\}## and ##W=\{(x,y,z)\in\mathbb{R}^3|x-y=0\}##. Find a matrix ##A## associated to a linear map ##f:\mathbb{R}^3\rightarrow\mathbb{R}^3## through the standard basis such that its nullspace is ##V##...

Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after show my...

Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after write my...

Homework Statement Build the matrix A associated with a linear transformation ƒ:ℝ3→ℝ3 that has the line x-4y=z=0 as its kernel. Homework Equations I don't see any relevant equation to be specified here . The Attempt at a Solution First of all, I tried to find a basis for the null space by...

Hey! :o I want to calculate the determinant of the following matrices: $$A=\begin{pmatrix}-3 & -11 & -11 & 45 \\ 1 & 11 & 10 & -83 \\ 1 & -6 & -5 & 81 \\ 0 & -3 & -3 & 42\end{pmatrix}$$ $$B=\begin{pmatrix}1+a_1 & a_2 & \ldots & a_n \\ a_1 & 1+a_2 & \ldots & a_n \\ \ldots & \ldots & \ldots...

I am currently brushing on my linear algebra skills when i read this For any Matrix A 1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out ) 2)(A + At)/2 is symmetric Now my question is , why should it be...

This is not part of my coursework but a question from a past paper (that we don't have solutions to). 1. Homework Statement Construct the matrix ##\sigma_{-} = \sigma_{x} - i\sigma_{y}## and show that the states resulting from ##\sigma_{-}## acting on the eigenstates of ##\sigma_{z} ## are...

Hi everyone I need raising and lowering operators for l=3 (so it should be 7 dimensional ). is it enough to use this formula: (J±)|j, m > =sqrt(j(j + 1) - m(m ± 1))|j, m ± 1 > The main problem is about calculating lx=2 for a given wave function , I know L^2 and Lz but I need L+ and L- to solve...

Hi PF! I have two matrices of equal dimensions. I would like to plot them such that row ##i## of matrix 1 corresponds the x-axis and row ##i## of matrix 2 corresponds to the y axis, so that row ##i## of matrix 1 and matrix 2 together give a coordinate. Does anyone know how to do this quickly...

Hey! :o Let $\mathbb{K}$ be a fiels and $A\in \mathbb{K}^{p\times q}$ and $B\in \mathbb{K}^{q\times r}$. I want to show that $\text{Rank}(AB)\leq \text{Rank}(A)$ and $\text{Rank}(AB)\leq \text{Rank}(B)$. We have that every column of $AB$ is a linear combination of the columns of $A$, or not...

So, we can break down the Dirac equation into 4 "component" equations for the wave function. I was going to post a question here a few days ago asking if a fermion (electron) could possesses a "spin" even if it were at rest, I.e., p=0. I did an internet scan, though, and found out that...

Homework Statement Happy new year all. I was wondering if you can use matrices to translate and transform a function? So for example if I were to take the function $$f(x)=x^2+4x$$ and I want to the translate and transform the equation to $$2f(x+4)$$. Can this be done by matrices. I know how...

I am given the following set of 4x4 matrices. How can i justify that they form a basis for the Lie Algebra of the group SO(4)? I know that they must be real matrices, and AA^{T}=\mathbb{I}, and the detA = +-1. Do i show that the matrices are linearly independent, verify these properties, and...

Covariant gamma matrices are defined by $$\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}=\{\gamma^{0},-\gamma^{1},-\gamma^{2},-\gamma^{3}\}.$$ -----------------------------------------------------------------------------------------------------------------------------------------------------------...

Hey! :o A set $R$ with two operations $+$ und $\cdot$ is a ring, if the following properties are satisfied: $(R, +)$ is a commutative/abelian group Associativity : For all $a,b, c \in R$ it holds that $(a \cdot b)\cdot c = a \cdot (b \cdot c)$. Distributive property : For all $a,b, c...

The gamma matrices ##\gamma^{\mu}## are defined by $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}.$$ --- There exist representations of the gamma matrices such as the Dirac basis and the Weyl basis. --- Is it possible to prove the relation...

The matrix representation ##U## for the group ##SU(2)## is given by ##U = \begin{bmatrix} \alpha & -\beta^{*} \\ \beta & \alpha^{*} \\ \end{bmatrix}## where ##\alpha## and ##\beta## are complex numbers and ##|\alpha|^{2}+|\beta|^{2}=1##. This can be derived using the unitary of...