Matrices Definition and 1000 Threads

Homework Statement This isn't a homework problem; it's just something I'm working on and I'm a little confused as to how to go about dealing with what I have. I have several traces of Dirac's gamma matrices, and I know that the trace of an odd number of gamma matrices is zero. So my first...

Homework Statement Homework EquationsThe Attempt at a Solution I need help with the second question, I did the first one correctly. My pre board is on Monday so please help.

Homework Statement Homework EquationsThe Attempt at a Solution No clue really. I went ahead and tried to simplify this by turnining it into an echelon matrix. But I am sort of stuck now, since I can't divide by -k because I don't know whether or not it is equal to 0?

Let $\mathbf{A} = \begin{pmatrix} 9 & 2 \\ 1 & 8 \end{pmatrix}.$ Obtain the general solution $\mathbf{y}(t)$ of the system of differential equations $\displaystyle \frac{d\mathbf{y}}{dt} = \mathbf{Ay}$: $\begin{cases} \frac{dy_1}{dt} = 9y_1+2y_2 \\ \frac{dy_2}{dt} = y_1+8y_2 \end{cases}$...

What's the dimension of the space of $2 \times 2$ matrices? What's the dimension of the space of $m \times n$ matrices? I know that matrices of size $m \times n$ with components in field $K$ form a vector space over $K$. To find the dimension, I would have to find basis. This I'm not quite sure...

Hello I was learning about determinants and matrices. I learned the generalization of getting the determinant of an n by n matrix. I then applied this to vector space (i + j + k) via a cross product and noticed that you leave the i j and k in their own columns in the first row of the matrix...

\Homework Statement Let B = ## \left[ \begin{array}{ccc} -1 & i & 1 \\ -i & 0 & 0 \\ 1 & 0 & 0 \end{array} \right] ##. Find a Unitary transformation to diagonalize B. Homework Equations N/A The Attempt at a Solution I have found both the Eigenvalues (0, 2, -1) and the Eigenvectors, which are...

Homework Statement Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question. Consider the...

Homework Statement Multiply these row excahnge matrices in the order pq qp and p^2 p = [0 1 0] [1 0 0] [0 0 1] q= [0 0 1] [0 1 0] [1 0 0] Homework EquationsThe Attempt at a Solution I don't understand why the solution is [0 1 0] [0 0 1] [1 0 0] do you not multiply rows by columns? When i...

Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...

Homework Statement Find the general solution Homework Equations Row operations Gaussian elimination The Attempt at a Solution This is has happened twice now and I'm not too sure how to deal with it. The last row ends up being all zeros except in that spot. I need to make this into an...

Homework Statement ##P=A(A^*A)^{-1}A^*## where A is a mxn real/complex matrix and ##A^*A## is invertible. ##A^*## means the conjugate transpose of A. Homework EquationsThe Attempt at a Solution Let y be in the range(A), such that ##y = Ax## for some ##x##. We can see that ##PA =...

Homework Statement find the general solution of the given system of equations: http://puu.sh/ncKaS/57a333f5b9.png Homework Equations Row Echelon Operations The Attempt at a Solution http://puu.sh/ncKcm/3e2b2bd5ab.jpg The correct answer given is x = 1, y = 1, z = 2, w = −3 I have done...

How do we get (6.265)? Shouldn't we have ##exp(-i\frac{\alpha}{2}\hat{n}.\sigma)=\cos(\frac{\alpha}{2}\hat{n}.\sigma)-i\sin(\frac{\alpha}{2}\hat{n}.\sigma)##?

I am having trouble with the following problem; a.) Find a matrix B in reduced echelon form such that B is row equivalent to the given matrix A. A=$$\left[\begin{array}{c}1 & 2 & 3 & -1 \\ 3 & 5 & 8 & -2 \\ 1 & 1 & 2 & 0 \end{array}\right]$$ So using my calculator I am able to get...

I am revising the basics of linear transformations and trying to get a thorough understanding of linear transformations and their matrices ... ... At present I am working through examples and exercises in Seymour Lipshutz' book: Linear Algebra, Fourth Edition (Schaum Series) ... ... At...

Firstly, my apologies to Deveno in the event that he has already answered these questions in a previous post ... Now ... Suppose we have a linear transformation $$T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2$$ , say ... Suppose also that $$\mathbb{R}^3$$ has basis $$B$$ and $$\mathbb{R}^2$$...

Homework Statement Suppose that AB = AC for matrices A, B, and C. Is it true that B must equal C? Prove the result or find a counterexample. Homework Equations Properties of matrix multiplication The Attempt at a Solution AC = A(D + B) = AD + AB = 0 + AB = AB ? Can someone help me...

Homework Statement Write in Vector-Matrix form then write the augmented matrix of the system. r + 2s + t = 1 r - 3s +3t = 1 4s - 5t = 3 Homework Equations The matrix to which the operations will be applied is called the augmented matrix of the system Ax = b, It is formed by appending the...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.8 The Dual of A Vector Space, Forms and Pullbacks ... I need help with...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.6 Constructing Linear Transformations ... I need help with a basic...

Homework Statement Calculate ##(\vec a \cdot \vec \sigma)^2##, ##(\vec a \cdot \vec \sigma)^3##, and ##(\vec a \cdot \vec \sigma)^4##, where ##\vec a## is a 3D-vector and ##\vec \sigma## is a 3D-vector formed from the ##\sigma_i## vectors. Homework Equations $$\sigma_1 = \begin{bmatrix} 0 &...

why are diagonal matrices and eigen vectors useful in vibrations analysis?

Hello! Okay- This is a relatively simple problem, but for some reason I'm having huge difficulty with it. So I have the equation of an ellipse, x^2-6sqrt3 * xy + 7y^2 =16, which I have converted into quadratic form to get (13, -3sqrt3, -sqrt3, 7) and I need to rotate it using the normal...

Due to the definition of spin-up (in my project ), \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} as opposed to \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} and the annihilation operator is...

I've been taught that for any system of linear equations, it has a corresponding matrix. Why do people sometimes use systems of linear equations to describe something and other times matrices? Is it all just a way of writing things down faster or are there things you could do to matrices that...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.2.9 (a) ... Exercise 1.2.9 (a) reads as follows:https://www.physicsforums.com/attachments/5101I am somewhat overwhelmed by this exercise ... can someone...

I don't really feel that I understand what it means for two matrices to be similar. Of course, I understand the need to understand ideas on their own terms, and that in math analogies are very much frowned upon. In asking if you know of any "reasonable" analogies for what it means for two...

I am beginning a study I have long wanted to engage in: quantum computing. This is a field lying at the intersection of mathematics, physics, computer science, and electrical engineering - all topics I studied, to varying levels. From time to time, I plan on posting notes and summaries that...

Homework Statement Given the 2 Matrices, find the transition matrices, Pd to c d = [ 1 1 0 ] [ 0 1 -1 ] [ 1 1 1 ] c= [ 1 0 0 ] [ 0 1 0 ] [ 0 0 1 ] Homework Equations Pd to c = [d] | [c] The Attempt at a Solution Pd to c = [d] | [c] [ 1 1 0 | 1 0 0 ] [ 0 1 -1 | 0 1 0 ]...

Homework Statement Hello guys; I am currently dealing with a problem that I have faced before several times and I would like to know a consistent way on how to solve it. I think what I want to do is diagonalize a matrix but I'm not sure if that's exactly it. Basically I have two or three...

Homework Statement If there are two sets of matrices ##S = \begin{Bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} | a, b, c, d \in \mathbb{C} \end{Bmatrix} ## and ##M = \begin{Bmatrix} \begin{bmatrix} a & b \\ -\overline{b} & \overline{a} \end{bmatrix} | a, b \in \mathbb{C} \wedge |a|...

Homework Statement Is there a non-degenerate 2x2 matrix that has only real eigenvalues but is not Hermitian? (Either find such a matrix, or prove that it doesn't exist) Homework EquationsThe Attempt at a Solution Here's my problem. I'm getting Contradicting results. So, I found this 2x2...

Homework Statement I'm using MATLAB to design a truss and the output is suppose to be the deflections caused by the applied loads. But I keep getting "not a number" for my deflections . here is the warnings I get. Warning: System may be partially constrained. > In truss3 (line 222) In...

Homework Statement How many hadamard matrices exists for size n? Homework Equations Hadamard matrices are square matrices whose entries are either +1 or −1 and whose rows are mutually orthogonal. The Attempt at a Solution I am just curious how many exists for 4, 8 and in general.[/B]

Hi, I was wondering if two matrices with the same eigenvalues share the same PDF. Any ideas and/or references would be helpful. Thanks in advance

Homework Statement There are three observers, all non accelerating. Observer B is moving at velocity vBA with respect to observer A. Observer C is moving at velocity vC B with respect to observer B. All three observers and all their relative velocities are directed along the same straight line...

Hi guys, iv been stuck on this problem for a while now and can't seem to make any headway construct a counterexample to the following statement: "for matrices A with real entries 'A^3=Identity impies A=Identity' im not restricted by size for the matrix. any hints would help because i just...

Homework Statement Consider hermitian matrices M1, M2, M3, M4 that obey the property Mi Mj + Mj Mi = 2δij I where I is the identity matrix and i,j=1,2,3,4 a) Show that the eigenvalues of Mi=+/- 1 (Hint: Go to the eigenbasis of Mi and use the equation for i=j) b) By considering the relation Mi...

Homework Statement Show that all ##n \times n## unitary matrices ##U## leave invariant the quadratic form ##|x_{1}|^{2} + |x_{2}|^{2} + \cdots + |x_{n}|^{2}##, that is, that if ##x'=Ux##, then ##|x'|^{2}=|x|^{2}##. Homework Equations The Attempt at a Solution ##|x'|^{2} = (x')^{\dagger}(x')...

Homework Statement Show that the set of all ##n \times n## orthogonal matrices forms a group. Homework Equations The Attempt at a Solution For two orthogonal matrices ##O_{1}## and ##O_{2}##, ##x'^{2} = x'^{T}x' = (O_{1}O_{2}x)^{T}(O_{1}O_{2}x) = x^{T}O_{2}^{T}O_{1}^{T}O_{1}O_{2}x =...

Trace of six gamma matrices I need to calculate this expression: $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$ I know that I can express this as: $$...

Homework Statement Show that all ##n \times n## (real) orthogonal matrices ##O## leave invariant the quadratic form ##x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}##, that is, that if ##x'=Ox##, then ##x'^{2}=x^{2}##. Homework Equations The Attempt at a Solution ##x'^{2} = (x')^{T}(x') =...

Homework Statement Show that no matrix A ∈ M3 (ℝ) exists so that A2 = -I3 Homework EquationsThe Attempt at a Solution This is from a french textbook of first year linear algebra. I'm quite familiar with properties of matrices but I don't have any idea of how to prove this. Thanks for the help!

Homework Statement A thin lens is placed 2m after the beam waist. The lens has f = 200mm. Find the appropriate system matrix. This is a past exam question I want to check I got right. Homework Equations For some straight section [[1 , d],[0 , 1]] and for a thin lens [[1 , 0],[-1/f , 1]]...

Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices. I want to prove that, but I haven't been able to. Please, could somebody point me a book where it is proven, or tell me how to...

Homework Statement Hi! I have the 3x3 matrix for L below, which I calculated. But now I need to figure out how the equation below actually means! Is it just the inverse of L (L^-1)? I cannot proceed if I don't know this step. Homework Equations See image The Attempt at a Solution I put in...

Light reflecting off a mirror actually penetrates a short distance into the mirror surface material. In metals, this distance is very short (much less than a wavelength) and so can be neglected. But metals tend to also absorb ~10% of the light, which is undesirable. Today’s modern multilayer...

So that's the question in the text. I having some issues I think with actually just comprehending what the question is asking me for. The texts answer is: all 3x3 matrices. My answer and reasoning is: the basis of the subspace of all rank 1 matrices is made up of the basis elements...

I think it's 3... All 2x2 can be written as a_1 A_1 + a_2 A_2 + a_3 A_3 + a_4 A_4 with A_1 = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} , A_2 = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} , A_3 = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} , A_4 = \begin{bmatrix} 0 & 0 \\ 0 & 1...