Matrices Definition and 1000 Threads

Is there a way without using the algorithm to find A-1 of a square matrix greater than 2x2? The question we are given in the books is: [-25 -9 -27] [536 185 537] [154 52 143] We are asked to find A-1 of the second and third column without computing the first column. (Sorry...

Why are the Pauli matrices multiplied by 1/2 ?? Why are they represented as σ1/2 σ2/2 and σ3/2 and not just σ1 σ2 σ3 ?

Hi everyone, I'm trying to find a general rule that expresses the product of two rotation matrices as a new matrix. I'm adopting the topological model of the rotation group, so any rotation which is specified by an angle \phi and an axis \hat{n} is written R(\hat{n}\phi)= R(\vec{\phi})...

Should the right answer to this question(below) be 14 and not 31? because A_{ij}^{k} means number of paths from i to j of length K. So A_{12}^{8} = 14 We then represent the graph as indcidence matrices and go from there on: A = { {0,1,0,0}, {1,0,1,0}, {1,1,0,1}, {1,0,0,0} } A[itex]^{8} = {...

Homework Statement A proof of equality between two traces of products of gamma matrices. Tr(\gamma^\mu (1_4-\gamma^5) A (1_4-\gamma^5) \gamma^\nu) = 2Tr(\gamma^\mu A (1_4-\gamma^5) \gamma^\nu) Where no special property of A is given, so we must assume it is just a random 4x4 matrix. 1_4...

Let A be the set of n \times n matrices. Then the identity element of this set under matrix multiplication is the identity matrix and it is unique. The proof follows from the monoidal properties of multiplication of square matrices. But if the matrix is not square, the left and right...

Homework Statement (1 1)^n = (1 n) (0 1) (0 1) Prove this through mathematical induction. Homework EquationsThe Attempt at a Solution I've replaced n with 1, so I've done that far. Then I said k = n. Then replaced all n with (k+1). I'm really stuck...

Homework Statement My instructor wants to know if there are finite or infinite amount of solutions Homework Equations Matrix Multiplication The Attempt at a Solution I pretty much turned A into a 3x3 matrix like this... | A11 A12 A13 | | A21 A22 A23 | | A31 A32 A33 | and...

How do we go about finding the transformation that was used to go from one matrix to another ( provided of course that the two are linked by a transformation) in general if all we have is two matrices.

Does anyone know where I can find or how I can compute (without checking all 512) the 8 diagonalizable 3x3 matrices over GF(2)? GF(2) means the entries are 0's and 1's. I'm working on some graph polynomial research and to check out a formula I'm working with I would have to take a sum over these...

Homework Statement See attached link The Attempt at a Solution I would have write my work out here, but I have not managed to display matrices next to each other yet. My problem is putting it into matrix form. The form I have is shown in the below attachement. The matrix I get for...

I just had my last Linear Algebra class, and I didn't get a chance to ask the one question that has been bugging me ever since we started in earnest with matrices. Why aren't there cube matrices? I mean, mathematical entities where numbers are 'laid out' in 3d not in 2d (not quite...

Homework Statement Let A be an 8x8 Boolean matrix. If the sum of A = 51, prove there is a row and a column such that when the total of the entries in the row and column are added, their sum is greater than 13. The Attempt at a Solution I considered a selection of one row and one column...

Show that all hermitian 2x2 matrices with trace 0 are elements of three dimensional vector space in \mathbb{R}, which basis vectors are Pauli spin matrices. Any clues on how to begin? :/

Homework Statement Homework Equations ImA = AIn = A. (A−1)−1 = A (AB)−1 = B−1A−1 The Attempt at a Solution Determinates: Det(A) = 3 – 0 = 3 Det (2A+BT) = 4 – 8 = -2 Matrices B^T = 2 -2 0 -5 (2A + B^T)^-1 = -8 -4 -8 -2 So I've kinda figured...

Hello MHB, Calculate $$A^{17}$$ where [FONT=Times]. Progress, I have multiplicate without adding them together to see a pattern and I can se at $$A^{17}$$ on that matrice where it's 6's it will be $$6^{17}$$ and rest I can't se any pattern those riight side of the triangle, cause the left will...

I'm trying to rotate a point about the origin (0,0,0) and starting with an identity matrix, this works fine for the x- and y-rotation axes, but fails with the z-axis, where the point just sits in place. \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} M_{ID} \times M_Z...

Homework Statement Suppose A and B are n × n matrices. Show that range(AB) ⊆ range(A) Homework Equations The Attempt at a Solution I think I need to show AB is a linear combination of the columns of A, but I'm not sure how to show this in general.

Fair to say there are "twice" as many square matrices as rectangular? Is it fair to say that there are at least twice as many square matrices as there are rectangular? I was thinking something like this... Let R be a rectangular matrix with m rows and n columns, and suppose either m < n...

I've been watching the OCW Linear Algebra course and I have a textbook, and I came across something that I think is fascinating: The Invertible Matrix Theorem. I've seen some proofs and I find that a lot of the statements are linked in different orders and sometimes the author will site one tiny...

Hi, First let's say you have an input vector and an output vector describing states. Let us use a binary number system, it doesn't really matter for mathematics. Say, I have an input vector V, components v_i. Now I would like to have an output vector W which can be of other dimension. The...

In almost all the books on field theory I've seen, the authors list out the different types of quantities you can construct from the Dirac spinors and the gamma matrices, but I'm confused by how these work. For instance, if $$\overline\psi\gamma^5\psi$$ is a pseudoscalar, how can...

Homework Statement Supposing P is a permutation matrix, I have to show that PT(I+P) = (I+P)T. Is there any general form of a permutation matrix I should use here as permutation matrices of a dimension can come in various forms. Homework Equations The Attempt at a Solution I did...

Homework Statement The problem: Attached as TheProblem.jpg. Solution: Homework Equations xT A x The Attempt at a Solution I computed the product and got 9x22 + 4 x1x1 + 4x12. I'm thinking that I might need to show that that obtained polynomial must be below zero since we want...

Question a) For each of the following matrices explain why the matrix is not invertible. i)[2,0,0_0,0,0_9,3,0] ii)[3,4,-6_7,2,1_-6,-812] iii)[5,-2,15_1,-4,3_2,1,6] b) Suppose A is an n×n matrix with the property that the equation Ax = 0 has only the trivial solution. Without using the...

Hello everyone, I’d like to find the following range equalities: Considering the following: A=B+C \\ A=B.C^T \\ A=[ B^T C^T ]^T I would like to find the function f for each equality above. .\\ dim( R(A) ) = f( R(B) , R(C) )\\ Considere that all matrices have compatible...

Let W1 = {A\in MnXn(R)| A = AT} and W2 = {A\in MnXn(R)| A = -AT} Show that MnXn = W1 (+) W2 where the definition of direct sum is: V is the direct sum of W1 and W2 in symbols: V = W1 (+) W2 if: V = W1 + W2 and W1 \cap W2 = {0} Attempt: I figure I have to show each...

It's my understanding that there is a direct correspondence between Schrodinger's wave equation and Heisenberg's matrix representations. I've always wanted to understand this equivalence but never really took the time to look into it. I'm just now getting back into re-learning Matrix or...

Here is the question: Here is a link to the question: Question on Similar Matrices? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

I'm learning about Matrix Theory/Linear Algebra and I find this to be pretty interesting in comparison to any other math (calc) I've learned so far. Do matrices get used much in physics?

Homework Statement What are the characteristics of a matrix that commutes with a matrix of ones? Homework Equations None. The Attempt at a Solution I'm helping a buddy with his homework and I can't figure out this problem.

In Jordan, Gauss-Jordan and Laplace it's necessary to miltiply a line by a constant, add the result to other line in order to obtain 'zeros' (to facilitate the process while using Laplace) or to obtain a identity matrix (Jordan and Gauss-Jordan). I take TOO long while doing this and sometimes...

Do you know where can I find proven identity det(AB)=det(A)det(B) using Levi Civita symbol.

I'm trying to figure out how to multiply and add square (3 X 3) matrices with Perl (without PDL) in which each matrix is in a separate file, but I haven't been making any progress. I'm a Perl beginner, so does anyone have any suggestions for examples/resources that may help me? Thanks.

Can we compare matrices? If A-B>0 is positive definite, can we say A>B?

Can we compare matrices? If A-B>0 is positive definite, can we say A>B?

Can anyone explain to a year 11 student what a dominant matrix is exactly? my textbook is not making much sense, i understand basic matricies and how you times them and rearange equations. Thank you so much(Happy)

1. The question Let V be a vector space with the ordered basis β={v1, v2,...,vn}. Define v0=0. Then there exists a linear transformation T:V→V such that T(vj) = vj+vj-1 for j=1,2,...,n. Compute [T]β. Homework Equations [T]γβ = (aij), 1≤i≤m, 1≤j≤n (where m is dimension of γ and n is the...

Homework Statement The problem is I am unable to understand the proof. I understand how it is done but I do not know how it is related to the theorem. It is probably because I am unable to understand the notation of matrices, the one involving k. It is given that I=δ_ij = 1 0 0...0...

I realize that Δ(s) is the cross product of the matrix on the left, but how did the solutions manual get the matrix on the far right multiplied by R_1(s) and R_2(s)? I need those matrix values to do the rest of the problem. Any help is appreciated, thank you.

Hi, Does anyone know how to prove that two commutative Hermitian matrices can always have the same set of eigenvectors? i.e. AB - BA=0 A and B are both Hermitian matrices, how to prove A and B have the same set of eigenvectors? Thanks!

Homework Statement Problem: Find and check the inverses (assuming they exist) of these three block matrices.: [1] {{I, 0},{C, I}} [2] {{A, 0}, {C, D}} [3] {{0, I}, {I, D}} Answer: [1] {{I, 0}, {-C, I}} [2] {{A^(-1), 0}, {-D^(-1) C A^(-1), D^(-1)}} [3] {{-D, I}, {I, 0}}...

Homework Statement Is U = {A| A \in nℝn, A is invertible} a subspace of nℝn, the space of all nxn matrices? The Attempt at a Solution This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and...

Homework Statement Suppose we know that a linear system Ax = b has a unique solution. What can we say about the solutions of the linear system Ax = 0? a) It has the same solution. b) The solution to Ax = b is also a solution to Ax=0, but there may be other solutions. c) Ax = 0 has a...

Homework Statement If A2 is a zero matrix, find all symmetric 2x2 nilpotent matrices. Homework Equations The Attempt at a Solution So if A2 is nilpotent, then [a,b;c,d]*[a,b;c,d] is equal to [0,0;0,0]. Since A is symmetric, b=c. Multiplying the two matrices, I get [ aa...

Hi, I am trying to find stationary points of the function f(x)=(xtAx)/(xtx) (the division of x transpose times A times x divided by x transpose x) where A is a px1 symmetric matrix. I need to take the derivative of this to show that when i set it to zero i get the eigenvectors of A. I know how...

Simple Matrices proof using Mathematica help! Homework Statement Hey guys, I'm trying to prove that (AB)-1 = B-1 A-1 and also the one that looks the same but is with transpose of the matrices making A and B arbitrary 3x3 matrices. I made A = {{a_1,a_2,a_3}...} B =...

Homework Statement Assuming that all matrices, A, B, C, and D, are n x n and invertible, solve for D. C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T} Homework Equations C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T} The Attempt at a Solution I must have missed something in...

I have no idea what this is! Please can someone explain comparing to a 3x3 matrix?

I am a sophomore majoring in Chemical Engineering. I have checked my classes requirement and seen that Chem E major doesn't need Matrices math or Quantum physics. I am wondering how so? And also, I am currently have 2 credits free for Spring semester. I am wondering should I study...