Matrices Definition and 1000 Threads

The lecturer said that a way to find the determinant of a matrix is to do the following det(A) = xdet(B) (1) where A is the original matrix, B is an arbirtray matrix and x is a scalar multiplier The lecturer also said that a simple way to find the determinant of a high...

Hello, I have a question. If A and B are NND matrices, how to prove C(A) belongs to C(A+B)? I can prove that C(A)<C(A,B) by using A=(A,B)transpose[(I,0)], and I also can prove C(A+B)<C(A,B) using the similar approach. But I cannot move further because my thoughs maybe not related to the...

Homework Statement Let T: P2 - P2 be the linear operator defined by T(a0 + a1x + a2x2) = a0 + a1(x - 1) + a2(x - 1)2 (a) Find the matrix for T with respect to the standard basis B = {1, x, x2}. Homework Equations [T]B[x]B = [T(x)]B The Attempt at a Solution T(1) = a0 + a1(1 -...

Homework Statement aij is a symmetric matrix bij is a an anti symmetric matrix prove that aij * bij = 0 Homework Equations aij * bij The Attempt at a Solution any one got any ideas ?

Hi, We know that the Pauli matrices along with the identity form a basis of 2x2 matrices. Any 2x2 matrix can be expressed as a linear combination of these four matrices. I know of one proof where I take a_{0}\sigma_{0}+a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{3}\sigma_{3}=0 Here, \sigma_{0} is...

Hi, After having solved some problems I encountered by using Google and often being linked to threads here, I finally decided to register, especially because I sometimes have problems for which I don't find solutions here and now want to ask them by myself :) Like the following: I am...

Would does it mean to say that two matrices or functions are orthogonal? What does this signify? I suppose it depends on the inner product. Say if the inner product is the trace of A(^T)B. Is there a real life application of orthogonal vectors in these sort of vector spaces?

Just a small question, I think I may have missed this part out in our lectures or something. :| Suppose I have a singular matrix A; will there always exist another matrix B such that AB (/BA) will be the zero matrix?

Hi, 1.Show that B satisfies the equation (B-pI)(B-qI) = 0 2.Hence, or otherwise, show that B-1 = 0.5(3I - B) In these kind of questions I don't know what they are testing me for! Let's take the first one as an example: The only skill they can possibly try to asses is whether I know how...

Homework Statement Homework Equations The Attempt at a Solution the usual method i.e. det(A - bI) = 0 i get the equation finally as b[3][/SUP] - 75b[2][/SUP] + 1850b -15576 = 0 from this i get b[1][/SUB][2][/SUP] + b[2][/SUB][2][/SUP] + b[3][/SUB][2][/SUP] = 1925 < 1949 is there an...

Homework Statement Let x = (x1,x2) \in Rn, x1 \in Rn1, x2 \in Rn2, n1 + n2 = n and A \in Rnxn be symmetric and positive definite. a) Let x0 \in Rn. Show that we can write (x-x0)TA(x-x0) = ||L(x-x0||22. Is L unique? b) Consider the quadratic term b = xTAx. Show that we can write b = x1TBx1 +...

Hi, I have: AX + X = B, all of them being matrices. I have the numbers in the A and B matrices and I have to find the exact values of a,b,c,d (numbers in the X matrix) I wanted to check if my method is correct: I multiplied both sides by A-1. 2X = A-1B So my values for abcd...

Homework Statement Let A and B be nxn matrices over reals. Show that I - BA is invertible if I - AB is invertible. Deduce that AB and BA have the same eigen values Homework Equations det(AB) = det(A).det(B) The Attempt at a Solution given: (I-AB) is invertible -> det(I-AB) is...

So my professor gave me an extra problem for Linear Algebra and I can't find anything about it in his lecture notes or textbooks or online. I think I've made it through some of the more difficult stuff, but I am running into a catch at the end. Homework Statement Find [;T(p(x))^{500};] when...

Homework Statement Let V be the spcae of all 3x3 matrices with real entries. Is W, the set of all 3x3 lower triangular matrices, a subspace of V? Why or why not? Homework Equations The Attempt at a Solution I just think that all 3x3 lower triangular matrices are included in...

Homework Statement Suppose V is a vector space. Is the set of all 2x2 invertible matrices closed under addition? If so, please prove it. If not, please provide a counter-example. Homework Equations The Attempt at a Solution well i know that what does it mean to be closed...

Homework Statement Find 2 more orthonormal polynomials on the interval [-2,1] up to degree 2 given that the first polynomial p(x) = 1/√3. ( Note: Take the highest coefficient to be positive and enter your answer as a decimal.) Homework Equations This is a web assign equation so the answer...

Homework Statement Divide A(x)= [x3+2x2+3 -4x3-x2-5] [3x2-2 x3-2x2+x+4] by B(x) = [x+4 -3] [-x+6 x+2] on both the right side and the left side. Homework Equations The Attempt at a Solution I am...

This is not a homework, only something embarrasing.. [T_8, T_4 + i T_5] = (3^(1/2) / 2) T_4 + i T_5 from http://phys.columbia.edu/~cyr/notes/QFT_3/lecture3.pdf" I can't see how to get the structure constant (3^(1/2) / 2). T_4 + i T_5 is a 3x3 matrix with a one at (2,3), the rest zeroes. I...

Given matrices in a vectorspace, how do you go about determining if they are independent or not? Since elements in a given vectorspace (like matrices) are vector elements of the space, I think we'd solve this the same way as we've solved for vectors in R1 -- c1u1 + c2u2 + c3u3 = 0. But I'm...

Homework Statement Given that \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}*1 where 1 is the identity matrix and the \gamma are the gamma matrices from the Dirac equation, prove that: \gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}*1 Homework Equations...

Hey all, I was hoping someone could explain to me how to calculate the angle between matrices, ie. two square matrices [ 2 0 0 -1] and [0 1 1 3^(1/2)] under the inner product <A|B> = trace (A^TB) Also, how would you go about determining an angle between...

Hi, I already posted this in solid state physics forum, but no one answered, so I guess this topic might belong to Mathematics. I read a text about crystallography where matrices were designated in the form: (S2 O S1) where S1 is input coordinate system, S2 is output coordinate...

Kindly ignore if some +- signs are placed wrongly in the equations. Thank you. Rotation in three dimensions can be represented using pauli matrices \sigma^{i}, by writing coordinates as X= x_{i}\sigma^{i}, and applying the transform X'= AXA^{-1}. Here A= I + n_{i}\sigma^{i}d\theta/2. The pauli...

Homework Statement Is it possible to solve this system of equations using matrices? x^2 + y^2 = 42 x+3y+2y^2=6 Homework Equations The Attempt at a Solution I solved the system of equations using the following MATLAB code. I'm kind of confused by the results. Are there two x values...

Can someone direct me to a good deep exposition of Jacobians and Hessians? I am especially looking for stuff that pertains to their being generalizations of derivatives of vector and scalar functions as well as div, grad, curl. Book sources or web links are appreciated.

I start to learn about matrices and their algebra, but I am wondering what physical application they have. I know that matrices have application in optics, which is called “Matrix Optics”, but do they have other applications? Can you give different and real physical examples with matrix algebra...

Hi :smile: I don't really know how to post a matrix here so I will try and make it as clear as possible. Matrix A: 4 -1 2 3 Matrix B: 6 4 -5 -3 Matrix C: 1 2 3 4 We are given that APB = C and we are asked for P What I did was I...

Homework Statement Prove the following theorem by induction: Let P be the transition matrix of a Markov chain. The ijth entry p(n)ij of the matrix Pn gives the probability that the Markov chain, starting in state si, will be in state sj after n steps. Homework Equations p(2)ij =...

Homework Statement Find all 2x2 square matrices A which are their own inverses. Homework Equations A2=I A=A-1 The Attempt at a Solution I know that the diagonal is comprised of 1s and or -1s and the other entries are zero but I can't seem to show it algebraically. I went the...

Hi, Given the two relations below, is it true and if yes, can anyone help me show that the solution to this must be the Pauli matrices? The alphas are matrices here. \alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i} = 2\delta_{ij}*1. 1 is the identity matrix \alpha_{i}^{2} = 1 Thank you

Homework Statement From Principles of Quantum Mechanics, 2nd edition by R Shankar, problem 1.8.10: By considering the commutator, show that the following Hermitian matrices may be simultaneously diagonalized. Find the eigenvectors common to both and verify that under a unitary transformation...

In Zee's Quantum Field theory book he writes \begin{align}\gamma^0 &= \begin{bmatrix}I & 0 \\ 0 & -I\end{bmatrix}=I \otimes \tau_3 \\ \gamma^i &= \begin{bmatrix}0 & \sigma^i \\ \sigma^i & 0\end{bmatrix}=\sigma^i \otimes \tau_2 \\ \gamma^5 &=\begin{bmatrix}0 & I \\ I & 0\end{bmatrix}=I \otimes...

Homework Statement What is the result of operating on the state |+> with the operator Sx? here, |+> denotes the eigenstate of Sz with eigenvalue 1/2. I am working in units where h-bar is 1 (for simplicity, and because I don't know how to type it) Homework Equations S_i = \frac{1}{2} σ_i The...

Homework Statement I'm trying to use the transfer matrix method in statistical mechanics but I'm struggling with the algebra so I'd like to know if there is a simpler way to find the eigenvalues and eigenvectors of a matrix. For example, studying the lattice gas model produces the transfer...

Hey guys There are those vectors made of Pauli matrices like \bar{\sigma}^\mu and {\sigma}^\mu. So if I have the product \bar{\sigma}^\mu {\sigma}^\nu I wonder if it is commutative? And if not, what is the commutator? Cheers, earth2

Let A = [a b; c d] a 2x2 matrix with complex entries. Suppose that A is row-reduced and also that a+b+c+d =0 . Prove that there are exactly three such matrices... so i realize that there are seven possible 2x2 matrices that are row-reduced. [1 0; 0 1], [0 1; 1 0], [0 0; 1 0], [0 0;0 1]...

If v is in Rn and is an eigenvector of matrix A, and P is an invertible matrix, how would you go about finding an eigenvector w of PAP-1? I'm thinking you have to use a fact about similarity?

I want to see if the matrix w = (1,0;0,1) is a linear combination of the matrices v1 = (1,2;-2,1) and v2 = (3,2;-1,1) where ; denotes a new line in the matrix. I know for example if w and v were 1xn matrices i.e vectors such as w = [1,1,1] v1= [2,-1,3] v2=[1,1,2] then i setup a matrix with...

I probably can remember the matrices by just trying to, but I hate having to "remember" things without actually understanding them. Is there no intuition behind these matrices so that I can remember it (the intuition) and then from it produce the wanted matrix? To me the matrices look like...

Hi all, Please help me to solve my problem in Mathematica that involves simultaneous equations with large matrices. The code is provided in the attached file, Exercise1.nb the errors say like this " General::unfl: Underflow occurred in computation. >> General::unfl: Underflow occurred...

Basically there are 2 equations ; x+2y+3z = 1 2x+4y+6z=2 I put them into a matrix and row reduce to get 1 2 3 | 1 0 0 0 | 0 so we can say x = 1 - 2y -3z and let y and z = 0 to get a solution is (1,0,0) Now i need to find the nullspace to find the whole solution set; so x +...

I also posted this in the homework help for introductory physics, but it wasn't getting any responses, so I guess it's slightly more advanced. Homework Statement Let L_b(a) denote the 4x4 matrix that gives a pure boost in the direction that makes an angle a with the x-axis in the xy plane...

Homework Statement If: A = l 1 -3 l , B = l -1 2 l C = l -12 -11 l l 2 1 l l 3 1 l l -10 -1 l Then, find X if AXB=C Homework Equations The Attempt at a Solution I don't know how I would start this problem??

Homework Statement If l -1 2 l l x l = l 2 l l 1 -2 l l y l l 1 l Show that the two lines are parallel and so never cross. Homework Equations The Attempt at a Solution I have attempted it, and so far all i have done is find the determentant. When i do this, i get...

Homework Statement Phil bought 4 meat pizzas, 4 veg pizzas and one loaf of garlic bread, and it costs him $92. By Abyy, who is partial to garlic bread, bough 2 meat pizzas, 2 veg pizzas and 10 loaves of garlic bread, and it cost $84. a) Use matrices to find Pmv, the most of 1 meat pizza...

Homework Statement Use Matrices to solve for x and y if: 2y - 4x - 5 = 0 and y = 3x + 1 Homework Equations The Attempt at a Solution I have done it, but i get a determinant of zero. So is this right? My working is the following: i rewrote both equations in the form of...

hi. I am trying to assign matrices to variables using for loop in MATLAB. can anyone help me wid dis? Thnx.

sorry I am new and posted instead of previewing...im currently writing the post

Is every complex symmetric (NOT unitary) matrix M diagonalizable in the form U^T M U, where U is a unitary matrix? Why?