Matrices Definition and 1000 Threads

Typically I understand that projection operators are defined as P_-=\frac{1}{2}(1-\gamma^5) P_+=\frac{1}{2}(1+\gamma^5) where typically also the fifth gamma matrices are defined as \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3 and.. as we choose different representations the projection...

What is the most convenient way to introduce matrices to students?

I have a problem regarding concatination of multiple matrices in MATLAB. For finite number of matrices there exists a command called cat or we may even put the matrices directly in a matrix representation format to get the desired concatenation. Like, for A and B to form a matrix C in...

Hi, Suppose we have the following matrix: \begin{center}\begin{pmatrix}\mathbf{L}&\mathbf{A}^T\\\mathbf{A}&\mathbf{0}\end{pmatrix}\end{center} where L is n-by-n matrix, A is m-by-n matrix. How to find the determinant of this square matrix? Thanks in advance

Please look at the link: http://gyazo.com/5f57caddf7dc76aa61d387f2915d29fe.png

Let L:P1 >> P1 be a linear transformation for which we know that L(t + 1) = 2t + 3 and L(t - 1) = 3t -2 a) Find L(6t-4) I just want to check the way to calculate this question. Is L(6t - 4) equal to 6*3t - 4*2 = 18t - 8? if not, how to calculate it?

given that 2 matrices have the same eigenvalues is it necessary that they be similar? If not, what is the connection between those 2?

I never really enjoyed learning the theory of maths and generally tried to avoid it at all costs since leaving University. However I'm looking at learning kinematic and dynamic chassis modelling and it requires extesive use of vectors and their transformations. I can follow the problems in...

Hey all, I have a quick question that should hopefully be simple to answer. Consider a the space of n \times n matrices over \mathbb C given by M_n(\mathbb C) . In order to properly consider this as a real matrix, we have to embed M_n(\mathbb C) \to M_{2n}(\mathbb R) , and I can give some...

Homework Statement Suppose that A and B are 5 x 5 matrices with the same Column Space (image). (a) Must they have the same columns? (b) Must they have the same rank? (c) Must they have kernels of the same dimension? (d) Must they have the same kernel? (e) If A is invertible, must B be...

Homework Statement Given a 4x5 matrix, what could a row of 0's represent geometrically? The Attempt at a Solution Given a scalar PLANE equation, you could make a 3x3 matrix and solve the system of equations. A row of zeroes there could represent a consisten but dependent...

Homework Statement Hi, I attached a file involving my problem and attempt at a solution.Thanks. Homework Equations The Attempt at a Solution

Homework Statement SUppose A and B are nxn matrices in the complex field and that A is unitarily similar to B. Homework Equations Show that Ak is unitarily similar to Bk for all k=1,2,3,.. The Attempt at a Solution I used induction to show its true for k=1 which it is. Then for...

Let A= |1 2| |0 3| and B= |1 0| |0 3| Show that A and B are not unitarily similar?

Basically, I need to solve a transient heat transfer conduction problem. I've got most of the work done but I need to solve the problem using MATLAB or C++ or some other kind of coding. That's what I need help with. The actual aspects of the problem aren't really that important, so I will try...

Homework Statement A and B are two matrices n X n Homework Equations AB-BA=identity matrix It is critical for me to prove that the are no matrices that are capable to hold the above equation true The Attempt at a SolutionI made several efforts.I had the idea to get the main...

Hi, How can I solve the equation below for M. G*inv(A+G'*inv(M)*G)*G'+F+M=0 G' is the transpose of G and inv(.) is the inverse of a matrix. Thanks

Homework Statement Suppose A is a normal matrix in the complex field.Homework Equations Show that ||Ax||=||A*x|| for all x in the complex fieldThe Attempt at a Solution If A is normal then AA*=A*A and ||Ax||=(Ax,Ax)=(x,A*Ax)

Homework Statement Given a matrix {latex] A_11 =A_22 = 0 A_12 =A_21 = x/y +y/x [ /latex] Find the contravariant components in polar coordinates. Answer: [itex] A_11 = 2 A_22 = -2/r^2 A_12 = 2cot(2 /theta)/r [ /latex] Homework Equations I used the polar coordinates metric to raise...

A set of m linear equations in n unknowns has the m × n matrix A of coefficients and the m × 1 (column) vector hT of right-hand sides. (Later we shall write this as AxT=hT). T = transpose In each of cases (a) to (d) below, answer as many as possible of the following...

Hi All, I recently came across the interesting notion of constructing the minimal set of nxn matrices that can be used as a basis to generate all nxn matrices given that matrix multiplication, and addition and multiplication by scalar are allowed. Is there a way to construct an explicit set...

Homework Statement Okay. So I have an equation: ABA + BAB = 2I where A and B are square nxn-matrices and I is the identity matrix. From this, I am supposed to find a way to express B as a function of A (given that A is close to I). So B = F(A), and it is also given that F(I) = I. Homework...

For a general, non-singular matrix A prove that A^-1=[(A^T A)^-1] A^T The Attempt at a Solution tried searching in textbook and internet-nothing yet someone somewhere must know an easy way to do this without having to sit there for five hours getting stuck but i just look at...

Hi, Everyone: I am reading a paper that refers to a "natural surjection" between M_g and the group of symplectic 2gx2g-matrices. All I know is this map is related to some action of M_g on H₁(S_g,Z). I think this action is...

I think I'm missing something real simple on trace theorems and Dirac matrices, but am just not seeing it. In the Peskin and Schroeder QFT text on page 135 we have: gamma^(mu)*gamma^(nu)*gamma_(mu) = -2*gamma^(nu) But, why can't we anti-commute and obtain the following...

Let T be the set of all matrics of the form AB - BA, where A and B are nxn matrics. Show that span T is not Mnn. 1) does "span T is not Mnn" mean that Mnn does not span T? Thanks

Can someone tell me (when RREFing a matrix) when do we put the vectors of a subspace in columns of a matrix and when in rows? Example from my notes: Here, my prof put the vectors in columns: and here, he put the vectors in rows: Thanks!

I get the idea of Jacobian matrices. I think. Working through different examples, I don't have any problems. For example, f1 = x^2 + y^2 f2 = 3x + 4y would result in [2x 2y] [3 4] Similarly, by my understanding, something like x^2 + y^2 3y + 4x would result in [2x...

Homework Statement Homework Equations The Attempt at a Solution I need help with (5) and (24). For (5), i can find MU and MV but have difficulty in finding MnU and MnV. For (24), i can solve (i) but don't know how to find (A+I)21B. The answer for (5): MnU=6nu; MnV=9nV... As for (24)...

I have a few questions relating to matrices. 1. All of the matrices I've worked with so far dealt with real numbers or real functions of real numbers. Can you work instead in complex numbers, and do you have to add or remove any rules because of this? 2. All of the matrices I've worked...

I've recently been learning about how to tell if a matrix is positive definite and how to create a positive definite matrix, but I haven't been given a reason why they're useful yet. I'm sure there are plenty of reasons, I just haven't seen them yet. In what ways do the properties of a positive...

Hey all, I have a question on this specific application of diagonalizable matrices. Homework Statement For what values of the real constant a is the matrix diagonalizable over \mathbb{C}? For what values is the matrix diagonalizable over \mathbb{R}? \begin{bmatrix} 0 & 0 & a\\ 1...

Q1. Find the value of a for which there are infinitely many solutions to the equations 2x + ay − z = 0 3x + 4y − (a + 1)z = 13 10x + 8y + (a − 4)z = 26 Now I know that for there to be infinitely many solutions the determinant of the coefficient matrix must = 0. I did this on a...

a)Find a basis for the space of 2x2 symmetric matrices. Prove that your answer is indeed a basis. b)Find the dimension of the space of n x n symmetric matrices. Justify your answer.

Homework Statement Show that a product of orthogonal matrices is orthogonal.Homework Equations Orthoganol matrix: M-1=MTThe Attempt at a Solution since A-1=AT A-1 and AT commute. commutable => symmetric => A-1AT=(A-1AT)T (A-1AT)-1=A-1AT/det(A-1AT) => (A-1AT)-1(A-1AT)=(A-1AT)2/det(A-1AT)...

Homework Statement If you have the matrix X = [ 1 1; 1 0] and Y = [ 0 -1; 1 0] and line segment A = {(0,y) | 0<y<1} Draw the images of line segment after you transform it by matrix X, Y, XY, YX (the image of A after transformation by linear transformation of L is {L(a) | all a in A} Homework...

Homework Statement Show that all 2 x 2 matrices with real entries: M(2x2) = { a b | a,b,c,d are real numbers} c d | is a vector space under the matrix addition: |a1 b1| + | a2 b2| = |a1+a2 b1+b2| |c1 d1| + | c2 d2| = |c1+c2 d1+d2| and scalar multiplication: r*| a b | = | ra...

Homework Statement Let A and B be 3x3 matrices over a field F. Prove that A and B are similar if and only if they have the same characteristic polynomial and the same minimal polynomial. Homework Equations The Attempt at a Solution

Hi, I need help with matrix derivation. I have 2 matrices of dimension 2x1, A and B. A = [f(x) x]^{T} B = [y x]^{T} I would like to find the dA/dB. How do I do this? and what is the dimension of the resultant matrix?

Hi, Everyone: In linear algebra courses, the defs/formulas for the sum, multiplication of matrices respectively, are often motivated by the fact that matrix addition models the point-wise addition of linear maps, i.e., If A,B are linear maps described on the same basis, then the sum...

I am studying the subject of linear dependence right now and had a question on this topic. Is it possible to construct a square matrix A such that the columns of A are linearly dependent, but the columns of the transpose of A are linearly independent? My intuition tells me no, but I'm not sure...

Hi all, I'd like to define a vector valued function in mathematica 7 as the exponential of a quadratic form, defined with respect to a purely symbolic matrix. What I want to do with it is to take derivatives with respect to the components of my vector, and evaluate the result when all...

1) Let A be an n x n matrix. Prove that if Ax= 0 for all n x 1matrices, then A=O. Can you show me the steps of solving this problem? Please!

I am curious how to derive the (I+N)^-1 = I - N + N^2 - N^3 + ... N^(k-1) + 0 Where N^k = O, because we assume that N is nilpotent. Actually I'm just supposed to show that the inverse always exists (for my homework), but I'm not asking how to find existence, I want to know how this equation...

Can anyone explain to me how to count the total # of non-invertible 2x2 matrices? I have the answer from the book, which is r^3+r^2-r provided r is a prime. But it doesn't explain how to get there, and I couldn't figure it out. I haven't been practicing linear algebra for quite a long...

The problem: I need to find the (minimal) rank of some matrix which is basically all parameters. For example, when i ask for the rank of \begin{pmatrix} a& b& c \\ d& e& f \\ g& h& i \end{pmatrix}, I get 3. I would like to get 1, since (excluding the possibility of a matrix of all 0's) it...

Homework Statement Consider a linear system of four equations with three unknowns. We are told that the system has a unique solution. What does the rref of the co efficient matrix look like? Homework Equations The Attempt at a Solution When it says "unique solution" I'm going to...

Homework Statement Find all solutions using Gauss-Jordan elimination: [ 0 0 0 1 2 -1 l 2 1 2 0 0 1 -1 l 0 1 2 2 0 -1 1 l 2] Homework Equations Switching rows, able to scale any row able to add non zero multiple to row The Attempt at a Solution What I did was...

Homework Statement A school has three clubs, and each student is required to belong to exactly one club. One year the students switched club membership as follows: Club A. 1/5 remain in A, 2/5 switch to B, and 2/5 switch to C. Club B. 1/4 remain in B, 1/2 switch to A, and 1/4 switch to...

You know the example "The space of functions from a set S to a field F" that's usually given in a linear algebra text? Well they never give an example of the set they're working in in detail so I defined the set as: ((S, (S x S, S, +)), ((F, (F x F, F, +')), (F x F, F, °)), (S x F, F, •))...