Matrices Definition and 1000 Threads

So I just started my Linear Algebra course yesterday. I am confused on one aspect. When asked to solve an augmented matrix, the teacher would employ row operations. I understand how the row operations lead from one matrix to the next, but what I don't understand is how we formulate which row...

Homework Statement A field is defined by f(i)=2i+1; f(j)=(j^2)-2 and f(k)=(-k^2)-2 a. Describe the Surface b. Find the gradient of the surface at the point This is the last question of an assignment but I actually have no dea on how to even start doing it. Also I've finished 2 assignments...

Hello, I guess this is a basic question. Let´s say that If I am given a matrix X it is possible to form a symmetric matrix by computing X+X^{T} . But how can I form a matrix which is both symmetric and orthogonal? That is: M=M^{T}=M^{-1}.

Homework Statement Let F(x)=(x-1)(x-2)(x-3)(x-4) be the charecteristic polynomial of A. Find the minimal polynomial of A^3 Homework Equations The Attempt at a Solution A is similar to a diagnol matrix with 1,2,3,4 on the diagnol, let's call it B. We know that A is diagnizable...

Verify that M(theta) is orthogonal, and find a unit vector n such that the line fixed by the reflection is given by the equation n . x = c, for a suitable constant c, which should also be determined. --------------- I did the verficiation part, by multiplying m(theta) by its...

Homework Statement Hi Is it correct that when I have a unitary 3x3 matrix U, then |Un,1|2+|Un,2|2+|Un,3|2=|U1,n|2+|U2,n|2+|U3,n|2, since UH=U? Here n denotes some integer between 1 and 3.

hi i got the eigen values as e=-1, e=i, -i as the imaginary roots and both 1 multiplicities can some one complete the question please thanks

Homework Statement A = floor(10*rand(6)) (6x6 matrix with random numbers) B = A'(transpose) A(:,6) = -sum(B(1:5,:))' (sum row 1st through 5th row entries and place in the 6th column and then transpose and take the negative) x = ones(6,1) (vector with 6 entries all equal to 1) Ax...

Let M_2x2 denote the space of 2x2 matrices with real coeffcients. Show that (a1 b1) . (a2 b2) (c1 d1) (c2 d2) = a1a2 + 2b1b2 + c1c2 + 2d1d2 defines an inner product on M_2x2. Find an orthogonal basis of the subspace S = (a b) such that a + 3b - c = 0 (c d) of M_2x2...

Consider the matrix A = u v^{\ast } where u, v \in \textbf{C}^{n}. Under what condition on u and v is A a projector?A is a projector if A^{2}=A , so we have u v^{*} u v^{*}= u v^{\ast }. Does this imply u v^{\ast } = I ? And what exactly are the conditions on u and v that they are asking? do...

Hi! I was taught that the dirac matrices are AT LEAST 4x4 matrices, so that means that I can find also matrices of higher dimensions. The question is: what do these higher-dimension-matrices represent? Are they just mathematical stuff or have they got a physical meaning? I ask that because in...

Homework Statement Let M1 = [1 1] and M2 = [-3 -2] ________[1 -1]_________[ 1 2] Consider the inner product <A,B> = trace(transpose(A)B) in the vector space R2x2 of 2x2 matrices. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R2x2 spanned by the...

I'm trying to show that the generators of the spinor representation: M^{\mu \nu}=\frac{1}{2}\sigma^{\mu \nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu] obey the Lorentz algebra: [M^{\mu \nu},M^{\rho \sigma}]=i(\delta^{\mu \rho}M^{\nu \sigma}-\delta^{\nu \rho}M^{\mu \sigma}+\delta^{\nu \sigma}M^{\mu...

Prove that the field R of real numbers is isomorphic to the ring of all 2 X 2 matrices of the form (0,0)(0,a), with a as an element of R. (Hint: Consider the function f given by f(a)=(0,0)(0,a).) I have no problem showing that it is a homomorphism & that it's injective. My question arrises...

Homework Statement Suppose N is an invertible n x n matrix, and let D = {f1, f2, ... , fn} where fi is column i of N for each i. If B is the standard basis of Rn, show that MBD(1Rn) = N. Call the standard basis of Rn = {E1, ... , En} Homework Equations The Attempt at a Solution The first...

Homework Statement Any spinor matrix can be expressed in a set of 16 linearly independent matrices. In the lecture the 16 Γ_J matrices (J=1 to 16) given are I, γ^0,1,2,3, σ^μυ, (γ^μ)(γ_5), iγ_5. I was asked to express M = (σ_μυ)(γ_5), (σ_μυ)(σ^μυ), (γ^α)(σ_μυ)(γ_α) in terms of the 16...

Homework Statement EDIT: There turns out to be a problem with the question, that's why it wasn't working. If anyone sees it they're just going to get confused, so I'm taking it off.Homework Equations The Attempt at a Solution

hello, if i have a sequence of definite positive matrices that converges, is it always that the limit matrix is always a definite positive matrix? if it's true, can someone please tell me why or link me to some proof? thank you.

Let A and B be 2 x 2 real matrices such that A = PBP^-1 for some invertible 2 x 2 complex matrix P. Prove that A = QBQ^-1 for some invertible 2 x 2 real matrix Q.

Let C be a 2 × 2 matrix such that x is an eigenvalue of C with multiplicity two and dimNul(C − xI) = 1. Prove that C = P |x 1|P^−1 |0 x| for some invertible 2 × 2 matrix P. I'm not sure where to start EDIT |x 1| |0 x| is the matrix I don't know why it's...

Vector and matrices...are the independent or dependent Homework Statement Determine whether the vectors v1=[1,-1;0,0],v2=[2,-2;1,... and v3=[-5,5;1,0] are independent or dependent. Find the span {v1,v2,v3} and give a description. Explain why W = [4,4;4,4,] is not in the span {v1,v2,v3}...

Homework Statement U is a unitary matrix. Show that ||UX|| = ||X|| for all X in the complex set. Also show that |λ| = 1 for every eigenvalue λ of U. Homework Equations The Attempt at a Solution I'm not sure where to start. So I looked up the definition of a unitary matrix. It...

Homework Statement Using u_k = \[ \left( \begin{array}{ccc} F_{k+1} \\ F_k \end{array} \right)\] u_0 = \[ \left( \begin{array}{ccc} 1 \\ 0 \end{array} \right)\] A = \[ \left( \begin{array}{ccc} 1 & 1 \\ 1 & 0 \end{array} \right)\] Solve for u_k in terms of u_0 to show that: F_k =...

S= \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix} for simple cubic I= \begin{bmatrix} -\frac{a}{2} & \frac{a}{2} & \frac{a}{2} \\ \frac{a}{2} & -\frac{a}{2} & \frac{a}{2} \\ \frac{a}{2} & \frac{a}{2} & -\frac{a}{2} \end{bmatrix}...

Homework Statement Hello, the problem is asking me to find a unitary matrix U such that (U bar)^T(H)(U) is diagonal. And we have H = [{7,2,0},{2,4,-2},{0,-2,5}] The Attempt at a Solution I don't know where to start. I tried getting the eigenvalues of the matrix A but that lead to...

Homework Statement I am trying to make sense of some of the terminology. Basically i get that each industry in an input matrix is dependant on each of those industries to some specified extent in order for production. Is the output what each sector produces before we take into acount...

If A and B are similar matrices, then show that A-xI and B-xI are similar were x is a scalar. How to start?

If A and B are similar matrices, show that Ak and Bk are similar. I am almost positive this has to be done by induction. p(k):= Bk=S-1*Ak*S p(k+1):= Bk+1=S-1*Ak+1*S Assume p(k) is true. I know I could take p(k) and multiply right by A but I don't think that will go any where.

Show that if A and B are similar matrices, then the det(A)=det(b). I am not entirely sure how to start this proof. I was thinking... A=ai j det(A)=\sumai j*(-1)i+j*Mi j from j=1 to n. I am pretty much clueless on this one though and not sure if I am just throwing stuff on the wall...

Homework Statement I'm trying to show that a given matrix is the inverse of the other, by showing that multiplying them together generates the identity matrix. I can't see a way to simplify the last step and I feel like I'm missing something..? Any input on this would be helpful. Thanks...

Homework Statement A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce: a) Find the inverse of the matrix and use it to solve the remaining...

So I have a couple of questions in regards to linear operators and their eigenvalues and how it relates to their matrices with respect to some basis. For example, I want to show that given a linear operator T such that T(x_1,x_2,x_3) = (3x_3, 2x_2, x_1) then T can be represented by a diagonal...

[URGENT] Rotation Matrices Homework Statement http://e.imagehost.org/0661/Screen_shot_2010-03-09_at_12_37_44_AM.png Homework Equations Rotation Matrix: cos(theta) -sin(theta) sin(theta) cos(theta) The Attempt at a Solution I understand 2a: cos(pi/4) -sin(pi/4) sin(pi/4)...

Homework Statement I'm supposed to derive the following: \left({\bf A} \cdot {\bf \sigma} \right) \left({\bf B }\cdot {\bf \sigma} \right) = {\bf A} \cdot {\bf B} I + i \left( {\bf A } \times {\bf B} \right) \cdot {\bf \sigma} using just the two following facts: Any 2x2 matrix can...

If S is subspace of R6x6 consisting of all lower triangular matrices, what is the dimension of S? Does anyone know the properties about dimension of lower triangular matrices?

Homework Statement Prove that if matrices A and b are similar and A is diagonalizable, then B is diagonalizable.Homework Equations this shows that A and B are similar i believe A = [P][B][P]^-1 and D = [P]^-1 [A] [P] means A is diagonalizable The Attempt at a Solution I believe this is a...

Homework Statement For l=1 the angular momentum components can be represented by the matrices: \hat{L_{x}} = \hbar \left[ \begin{array}{ccc} 0 & \sqrt{\frac{1}{2}} & 0 \\ \sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} \\ 0 & \sqrt{\frac{1}{2}} & 0 \end{array} \right] \hat{L_{y}} =...

Homework Statement For the invertible matrices A, B and A-B, simplify the expression (A - B)^{-1}A(A^{-1} - B^{-1})B. Homework Equations properties of invertible matrices The Attempt at a Solution (A - B)^{-1}A(A^{-1} - B^{-1})B = (A - B)^{-1}(AA^{-1}B - AB^{-1}B) = (A -...

Can someone help me solve this problem using matrices and not Kirchhoff's rules. My professor posted this as a challenge question and I would like to know how to solve such a problem. Please help! I tried looking up how to solve this type of problem using matrices but to no luck. I don't...

Homework Statement Prove that if two matrices are similar then they have the same eigenvalues with the same algebraic and geometric multiplicity. Homework Equations Matrices A,B are similar if A = C\breve{}BC for some invertible C (and C inverse is denoted C\breve{} because I tried for a...

e11, e12, e21, and e22 are 2x2 matrices with all zeros except for a one in ij=11, ij =12, ij=21, and ij=22, respectively. I know how to show if vectors are a spanning set but I don't know what to do about matrices. Can someone show me how to approach this? Thanks.

If I have two matrices A and B and I want to show they are similar, is it enough to show that Trace(A)=Trace(B) or instead show that Det(A)=Det(B)? Thanks Tal

Inverting big matrices. REALLY BIG! How is it done? Let's say I have a sparsely populated 1 gazillion by 1 gazillion square matrix in a formula like this A*x = b. What sort of efficient methods exist to do the following?: find the rank, invert it if it has full rank, find the null vectors...

I'm trying to understand the Wiener Filter, and I have a few questions. 1. How can there be such a thing as a correlation matrix of 1 vector. I read here: R_yy = E[ y(k) * y^T(k) ] where y(k) is a vector, and y_T(x) is the same vector transposed. I thought correlation represents the...

Homework Statement Prove that similar matrices have the same rank. Homework Equations The Attempt at a Solution Similar matrices are related via: B = P-1AP, where B, A and P are nxn matrices.. since P is invertible, it rank(P) = n, and so since the main diagonal of P all > 0...

Homework Statement If A and B are invertible matrices and B is similar to A, prove that B-1 is similar to A-1 Homework Equations The Attempt at a Solution Not sure how to do this.. I know that similar matrices have the same characteristic polynomials and the same eigenvalues and...

Homework Statement For each matrix A below, let T be the linear operator on R3 thathas matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. Find the algebraic and geometric multiplicities of each eigenvalues, and a basis for each eigenspace. a) A =...

I am reading Munkres and know exactly how to find the fundamental groups of surfaces, using pi_1 and reducing it down to simpler problems. However, I'm completely lost when looking at my final exam it says to find the fundamental groups of matrices! How do you go about doing that! There are...

Hi guys I have been sitting here for a while thinking of why it is that for a unitary matrix U we have that UijU*ji = |Uij|2. What property of unitary matrices is it that gives U this property? Niles.

Hi My QFT course assumes the following notation for gamma matrices: \gamma ^{\mu_1 \mu_2 \mu_3 \mu_4} = {\gamma ^ {[\mu_1}}{\gamma ^ {\mu_2}}{\gamma ^ {\mu_3}}{\gamma ^ {\mu_4 ]}} what does the thing on the right hand side actually mean? Its seems to be a commutator of some sort.