In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.
These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field.
Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices.
This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.
Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space R3.
The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for the real Lie algebra
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
, which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of R3, and the (unital associative) algebra generated by iσ1, iσ2, iσ3 is isomorphic to that of quaternions.
Homework Statement
A=[a,1;0,a] B=[a,0;0,a]
If I want to show if matrix A is NOT similar to matrix B. Is it enough to show that B=/=Inv(P)*A*P? Or would I need to show that they do not have both the same eigenvalues and corresponding eigenvectors?
Hello. I need some help with one question about relationship of two matrices.
The task:
Suppose that I is identity matrix, u - is vector, u' is transposed vector, α - real number. It can be prove that inverse matrix of I+α*u*u' has similar form I+x*u*u'. The task is to find x.
I tried to...
Just trying to get my head around undergraduate quantum mechanics and they throw a lot of stuff at you, so some help is appreciated.
I understand that the wave function is some abstract vector living in an infinite dimensional hilbert space, and that it's a function. But then the textbook I'm...
How does one construct a block diagram from the state space representation?
Consider the state space:
\begin{align}
\dot{\mathbf{x}} &=
\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}
\mathbf{x}(t) +
\begin{pmatrix}
0\\
1
\end{pmatrix}\mathbf{u}(t)\\
y(t) &= x_1(t)
\end{align}
Is possible to rewrite the quadratic, cubic and quartic determinant in terms of matrices and matrix operations (trace and determinant)?
https://en.wikipedia.org/wiki/Discriminant_of_a_polynomial#Formulas_for_low_degrees
Homework Statement
The Attempt at a Solution
I know what relations are individually but what do I do to represent the composition of both? Is it some matrix operation? Would I multiply them, but instead of adding I use the boolean sum?
Homework Statement
If A and B are invertible matrices over an algebraically closed field k , show there exists \lambda \in k such that det(\lambda A + B) = 0 .The Attempt at a Solution
Can anyone agree with the following short proof? I tried looking online for a confirmation, but I wasn't...
Homework Statement
This makes intuitive sense to me, but I am getting stuck when trying to read the Dirac notation proof.
Anyway, the author (Shankar) is just demonstrating that the product of two operators is equal to the product of the matrices representing the factors.
Homework Equations...
1) Let A a square matrix, x a colum vector and b another colum vector. So, I want solve for x the following equation: Ax=b
So: x=b÷A = b×A-1 And this is the answer! Or would be this the correct answer x = A-1×b ?
2) Is possible to solve the equation above for A ? How?
First off, I apologize if I'm in the wrong thread. I wasn't really sure where to put this.
Alright, long winded question so stay with me (note: the actual question is at the end, so if you already know how to work it out, just skip ahead)! I was reading a math problem at the nsa.gov website (...
So let's say I have 2 matrices A and B. I need to solve 2 eqns involving specific elements of each matrix.
e.g. A(1)+B(2)=4; A(1)-B(2)=2.
Is there any way to do this? My efforts with Fsolve and solve have failed.
Here's what I've done so far:
function F=myfun(A,B)...
Homework Statement
a) Find the inverse of the matrix:
\begin{pmatrix}1 & 2 & 0\\
2 & 0 & 1\\
1 & 1 & 2\end{pmatrix}
(sorry I don't know how to show a matrix more clearly on this)
b) Write A and A-1 as a composition of matrices of the form Rij(k), Tij and D22(k)
Homework...
Homework Statement
I thought it would be better to attach it.
Homework Equations
The Attempt at a Solution
So for the first part I've found that A^2=the Identity matrix, but from there I don't have much of an idea on how to substitute that into the equation for M and end up with...
Hello,
The product of a 2x5 matrix P and a 5x3 matrix B shall be a 2x3 zero matrix. P and B are all matrices of integers.
P = [6 2 -5 -6 1;3 6 1 -6 -5]
One possible B is [0 -4 0;3 0 0;-1 -1 3;2 -3 -2;1 1 3]
This solution B has a property: det(PPt) = det(BtB) = 7778
The question is: What...
Homework Statement
A polynomial of degree two or less can be written on the form p(x) = a0 + a1x + a2x2.
In standard basis {1, x, x2} the coordinates becomes p(x) = a0 + a1x + a2x2 equivalent to ##[p(x)]_s=\begin{pmatrix}a0\\ a1\\ a2 \end{pmatrix}##.
Part a)
If we replace x with...
I am trying to find the state equations for a mass spring system.
I found the transfer function to be
\[
H(s) = \frac{X_1(s)}{F(s)} = \frac{m_2s^2 + b_2s + k}
{s\big[m_1m_2s^3 + (m_2b_1 + m_1b_2)s^2 +...
One last question on these topics, I need to choose the WRONG statement, and they all seem correct to me...
a) If A is a squared matrix for which
\[A^{2}-A=0\]
then A=0 or A=i
b) If A and B are diagonal matrices, then Ab=BA
c) A 4X4 matrix with eigenvalues 1,0,-1,2 is "diagonlizable"
d) The...
Homework Statement
Let A be the set of all 3x3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices B in A for which the system of linear equations B \left[ \begin{array}{c} x \\ y \\ z \end{array} \right] =...
Hey folks.
I'm working on a project which seems to be encountering a problem.
I took Linear Algebra a few years ago in college, and haven't really applied it very much so I'm at a bit of a loss here.
I have a solution to a iterative nonlinear least squares problem: Trilateration with n...
I'm trying to find a general solution for the logistic ODE \frac{dU}{dx}=A(I-U)U, where A and U are square matrices and x is a scalar parameter. Inspired by the scalar equivalent I guessed that U=(I+e^{-Ax})^{-1} is a valid solution; however, U=(I+e^{-Ax+B})^{-1} is not when U and A don't...
Hi all,
I have two matrices
A=0 0 1 0
0 0 0 1
a b a b
c d c d
and B=0 0 0 0
0 0 0 0
0 0 a b
0 0 c d
I need to prove that two eigenvalues of A and two eigenvalues of B are equal. I tried to take the determinant of A-λI...
Could anyone let me know of a good resource that I could use to learn matrix mathematics? I'm not sure if that is the proper term for that segment of mathematics, but hopefully you get the gist of it. It can be a book or a website, does not matter to me. Also, any suggestions as to what I...
I need to prove that a 3x3 matrix with all odd entries will have a determinant that is a multiple of 4.
This is how I set it up:
I let A = { {a, b, c}, {d, e, f}, {g, h, i} } with all odd entries
then I define B = { {a, b, c}, {d + na, e + nb, f + nc}, {g + ma, h + bm, i + cm} }
where I add...
Homework Statement
2. The attempt at a solution
So part a. makes sense to me, it basically comes down to
A1 =
1 -1 -1
-1 1 1
-1 1 1
A2 =
1 -1 -1
-1 2 -2
-1 -2 11
I'm not sure how to approach part b. because the question doesn't make much sense to me...
Matrix proof :(
1. Let A and B be two similar matrices.
characteristic in the space λ is an eigen value,
show that : sized V_λ^A = sized V_λ^B
2. Let A invertible matrix.
A ∈ ℝ nxn and invertible matrix ⇔ 0, A is not an eigen value.
3. Let A and B be two similar matrices...
##GL(n,\mathbb{R})## is set of invertible matrices with real entries. We know that
SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R})
is there any specific subgroups of ##GL(n,\mathbb{R})## that is highly important.
Hi everyone, :)
Here's is a question I have trouble understanding. Hope you can help me out. :) Specifically what is meant by the structure tensor and how is it computed when given a \(2\times 2\) triangular matrix?
Problem:
Write the structure tensor for the algebra \(A\) of traingular...
Hello
I have been trying to solve a couple of true / false questions, and I am not sure my answers are correct, I would appreciate it if you could verify it.
The first question is:
A and B are matrices such that it is possible to calculate:
\[C=AB+B^{t}A^{t}\]
a. A and B are of the same...
An operator A defined by a matrix can be written as something like:
A = Ʃi,jlei><ejl <eilAlej>
How does this representation translate to a continuous basis, e.g. position basis, where operators are not matrices but rather differential operators etc. Can we still write for e.g. the kinetic...
Lets suppose a 4×4 matrix A has two identical rows with some other 4×4 matrix B. Does that imply there determinant is equal? Or does it really say nothing about how the determinants of the two matrices are related.
Hi,
In QFT we define the projection operators:
\begin{equation}
P_{\pm} = \frac{1}{2} ( 1 \pm \gamma^5)
\end{equation}
and define the left- and right-handed parts of the Dirac spinor as:
\begin{align}
\psi_R & = P_+ \psi \\
\psi_L & = P_- \psi
\end{align}
I was wondering if the left- and...
As far as I know there is no explicit formulas but is this true? I've tested it in Matlab with random matrices and It seems true!
cond(A+B) =< cond(A) + cond(B)
Where can I find a proof for this hypothesis?
Dear PhysicsForum,
We have just treated the Dirac equation and its lagrangian during our QFT course, but we have only gone in depth in 3+1 dimensions.
My question is about what happens to spin in 2+1 dimensions. In 3+1 dimensions we have to use 4 by 4 gamma matrices, but in 2+1 dimensions we...
Homework Statement
Whats up guys!
I've got this question typed up in Word cos I reckon its faster:
http://imageshack.com/a/img5/2286/br30.jpg
Homework Equations
I don't know of any
The Attempt at a Solution
I don't know where to start! can u guys help me out please?
Thanks!
Hi everyone, :)
I have very limited knowledge on linear algebra and things like Jordan Normal form of matrices. However I am currently doing an Advanced Linear Algebra course which is compulsory and I am trying hard to understand the content which is quite difficult for me. One of the things...
Please do not be offended by my literary style. I find thinking about mathematical problems in such a way helps me learn better.
A is a 2x2 matrix of complex numbers, call this "apple"
B is a 2x2 matrix of complex numbers, call this "banana"
Let a "Fruit Salad" be defined as follows:
S...
I need to determine the matrix that represents the following rotation of $R^3$.
(a) angle $\theta$, the axis $e_2$
(b) angle $2\pi/3$, axis contains the vector $(1,1,1)^t$
(c) angle $\pi/2$, axis contains the vector $(1,1,0)^t$
Now, I would like to check if I got the right answers because...
1. Consider the 2x2 matrix \sigma^{\mu}=(1,\sigma_{i}) where \sigma^{\mu}=(1,\sigma)
where 1 is the identity matrix and \sigma_{i} the pauli matrices. Show with a direct calcuation that detX=x^{\mu}x_{\mu}
3. I'm not sure how to attempt this at all...
I posted this question over at the QM page,
https://www.physicsforums.com/showthread.php?t=714076
but I realized I am really looking for a
hard Mathematical proof ...
A description of a numerical way of proving this would also be very helpful for me.
or a reference covering the...
Just going over my linear algebra notes and I've forgotten the formal definition of ## \epsilon(i,j)_{rs} ##
I have written down ## \epsilon (i,j)_{rs} = \delta_{ir}\delta_{js} ## but I can't seem to remember what r and s represent. Also, I don't quite understand why it equals ##...
Hi,
Wasn't sure if I should post this to Linear Algebra or here.
My question is really simple:
Can a 2N by 2N random, and Hermitian Matrix ( Hamiltonian ) be always written as:
H = A \otimes I_{2\times 2} + B \otimes \sigma_x + C \otimes \sigma_y + D \otimes \sigma_z
where A,B,C,D are all...
True or False. If true explain or prove answer, and if false give an example to show the statement is not always true.
1. If A is a 4x4 matrix and a1+a2=a3+2a4, then A must be singular.
2. If A is row equivalent to both B and C, then A is row equivalent to B+C.
My Work:
1. I say it's...
Homework Statement
Find all the values of h and k such that the system:
hx + 6y = 2
x + (h+1)y = k
has: (a) No solutions (b) A unique solution (c) Infinitely many solutions
Homework Equations
-
The Attempt at a Solution
I've tried putting the system into echelon form and got...
Homework Statement
Here is the problem:
http://img801.imageshack.us/img801/6770/oaza.png
Homework Equations
None really, just gauss jordon elimination I assume unless I am missing out on something
The Attempt at a Solution
First I multiplied the first row by -5 then added...
Hi everyone,
I have two matrices A and B,
A=[0 0 1 0; 0 0 0 1; a b a b; c d c d] and B=[0 0 0 0; 0 0 0 0; 0 0 a b; 0 0 c d].
I have to proves theoretically that two of the eigenvalues of A and B are equal and remaining two eigenvalues of A are 1,1.
I tried it by calculating the...
I read that scalar matrices are the center of the ring of matrices. How would I prove this?
Tips are appreciated. It is already obvious that scalar matrices commute with all matrices, but the converse seems tricky.
BiP