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.
Hi Everyone.
Just a brief hello before the problem! I am a new user as of today. I am studying Electrical Engineering in my spare time after work, and currently working full time an electronics service engineer. I have tried to make the problem as clear as I can, any help would be highly...
Hello all,
I have 3 matrices, A - symmetric, B - anti symmetric, and P - any matrix
All matrices are of order nXn and are not the 0 matrix
I need to tell if the following matrices are symmetric or anti symmetric:
1) 5AB-5BA
2) 4B^3
3) A(P^t)(A^t)
4) (A+B)^2
5) BAB
How would you approach...
Homework Statement
Given the ellipse
##0.084x^2 − 0.079xy + 0.107y^2 = 1 ##
Find the semi-major and semi-minor axes of this ellipse, and a unit vector in the
direction of each axis.
I have calculated the semi-major and minor axes, I am just stuck on the final part.
Homework Equations
this...
I've attached the question to this post. The answer is false but why is it not considered the orthogonal projection?
##
A = \begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}
##
##
B = \begin{bmatrix}
x \\
y
\end{bmatrix}
##
##
AB =...
I'm trying to understand what makes a valid covariance matrix valid. Wikipedia tells me all covariance matrices are positive semidefinite (and, in fact, they're positive definite unless one signal is an exact linear combination of others). I don't have a very good idea of what this means in...
Hello,
I am currently trying to study the mathematics of quantum mechanics. Today I cam across the theorem that says that a Hermitian matrix of dimensionality ##n## will always have ##n## independent eigenvectors/eigenvalues. And my goal is to prove this. I haven't taken any linear algebra...
Two questions, both about adjacency matrices (graphs). The first, specific, the second, general.
[1] I read:
"Consider a directed graph and a positive integer k. Then the number of directed walks from node i to node j of length k is the entry on row i and column j of the matrix Ak..." [where A...
OK so i can prove that the given inverse is actually the inverse but i can not prove that I+A is non singular without using the given inverse so how do i go about doing that?(I have done part a)
Thanks in advance.
Find all values of h and k such that the system has no solutions, a unique solution, infinitely many solutions
hx +6y =2
x (h+1)y =2k
I can't seem to augment the matrix. Am I allowed to multiply buy variables h / k?
I can find the determinant: h^2 +h -6
Then make it equal to 0 and solve; h =...
HI, I've been running through my lectures notes and have stumbled upon something i can't quite figure out.
I am given
Ψ(x)=∑a_iΨ_i(x)
Then
OΨ(x)=∑ a_i O Ψ_i(x) , where O is an operator acting upon Ψ
Then i am given something which i don't quite understand,
OΨ_i(x) = ∑ O_ji Ψ_j(x) , Where...
Homework Statement
Generate a triangle. For this problem, generate a triangle at a grid of points that are finely spaced in the x dimension. The triangle is defined as follows:
-Side 1: y = 0 for x = 0 to 2
-Side 2: x = 0 for y = 0 to 1
-Hypotenuse: y = 1-0.5x for x = 0 to 2
Alternatively, the...
Mod note: Moved from a technical section, so is missing the homework template.
I am using matrix methods to do ray optics but my knowledge on matrices is behind.
I found the system matrix to be
\begin{bmatrix} \frac{-f_2}{f_1} & f_1 + f_2 \\ 0 & \frac{-f_1}{f_2} \end{bmatrix}
I want to find...
Homework Statement
The problem states that we have L as the linear transformation as:
\begin{align*}
A=
\left(
\begin{array}{ccc}
2 & 0 & 1 \\
-2 & 3 & 2 \\
4 & 1 & 5
\end{array}
\right)
\end{align*}
And when given another linear transformation T as:
\begin{align*}
B=
\left(
\begin{array}{ccc}...
I know that the matrices {\Gamma^{A}} obey the trace orthogonality relation Tr(\Gamma^{A}\Gamma_{B})=2^{m}\delta^{A}_{B}
In order to show that a matrix M can be expanded in the basis \Gamma^{A} in the following way
M=\sum_{A}m_{A}\Gamma^{A}
m_{A}=\frac{1}{2^{m}}Tr(M\Gamma_{A})
is it enough to...
Homework Statement
Suppose a 2x2 matrix X (not necessarily hermitian, nor unitary) is written as
X = a0 + sigma . a (the sigma . a is a dot product between sigma and a)
where a0 and a1, a2 and a3 are numbers.
How on Earth does X represent a matrix? it's a number added to another number...
Hey PF, I'm having trouble seeing the bigger picture here.
Take matrix A and matrix B. If B can be obtained from A by elementary row operations then the two matrices are row equivalent. The only explanation my book gives is that since B was obtained by elementary row operations, (scalar...
Suppose we have linear operators A' and B'. We define their sum C'=A'+B' such that
C'|v>=(A'+B')|v>=A'|v>+B'|v>.
Now we can represent A',B',C' by matrices A,B,C respectively. I have a question about proving that if C'=A'+B', C=A+B holds. The proof is
Using the above with Einstein summation...
Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the...
Homework Statement
[/B]
Let Z be any 3×3 orthogonal matrix and let A = Z-1DZ where D is a diagonal matrix with positive integers along its diagonal.
Show that the product <x, y> A = x · Ay is an inner product for R3.
Homework Equations
None
The Attempt at a Solution
I've shown that x · Dy is...
Homework Statement
Suppose that ##s \to A(s) \subset \mathbb{M}_{33}(\mathbb{R})## is smooth and that ##A(s)## is antisymmetric for all ##s##. If ##Q_0 \in SO(3)##, show that the unique solution (which you may assume exists) to
$$\dot{Q}(s) = A(s)Q(s), \quad Q(0) = Q_0$$
satisfies ##Q(s) \in...
Homework Statement
Hello!
Please, take a look at the problem described in the attached file.
The question is: Explain why the transition matrix does what we want it to do.Homework Equations
The Attempt at a Solution
(sorry, I don't know yet how to type formulas)
I don't quite understand this...
Homework Statement
Hello!
Here is the problem from Stitz-Zeager Pre-calculus book:
At 9 PM, the temperature was 60F; at midnight, the temperature was 50F; and at 6 AM,
the temperature was 70F . Use the technique in Example 8.2.3 to t a quadratic function
to these data with the temperature...
Homework Statement
I need to show that any normal matrix can be expressed as the sum of two commuting self adjoint matrices
Homework Equations
Normal matrix A: [A,A^\dagger]=0
Self Adjoint matrix: B=B^\dagger
The Attempt at a Solution
A is a normal matrix. I assume I can write...
This is a pretty basic question, but I haven't seen it dealt with in the texts that I have used. In the proof where it is shown that the product of a spinor and its Dirac conjugate is Lorentz invariant, it is assumed that the gamma matrix \gamma^0 is invariant under a Lorentz transformation. I...
I am spending time revising vector spaces. I am using Dummit and Foote: Abstract Algebra (Chapter 11) and also the book Linear Algebra by Stephen Freidberg, Arnold Insel and Lawrence Spence.
On page 419 D&F define similar matrices as follows:
They then state the following:
BUT? ... how...
Hi All struggling with concepts involved here
So I have {P}_{2} = \left\{ a{t}^{2}+bt+c \mid a,b,c\epsilon R\right\} is a real vector space with respect to the usual addition of polynomials and multiplication of a polynomial by a constant.
I need to show that both...
Right, i don't believe this is a homework question. The only reason I am stating this is because PF are stringent with their rules.
I'm quite confused and I'm not sure how to explicitly state my problem.
The vertices of a triangle are (a,b) (c,d) and (e,f).
This can be arranged into a...
Homework Statement
Let R be the ring of all 2*2 matrices over Zp, p a prime,. Show that if det(a b c d) = ad - bc ≠ 0, then (a b c d) is invertible in R.Homework Equations
The Attempt at a Solution
I don't know how to start if Zp, with p a prime, is the clause. I know that since ad- bc ≠ 0, it...
Homework Statement
Find the matrix representation of S_z in the S_x basis for spin 1/2.
Homework Equations
I have the Pauli matrices, and I also have the respective kets derived in each basis. There aren't really any relevant equations, other than the eigenvalue equations for the...
x=t^2-s^2, y=ts,u=x,v=-y
a) compute derivative matrices \vec{D}f(x,y) = \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right]
\vec{D}f(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right]
b) express (u,v) in terms of (t,s)
f(u(x,y),v(x,y) = (t^2-s^2,-(ts))
c) Evaluate \vec{D}(u,v)...
The problem statement
Let ##A ∈ K^{m×n}## and ##B ∈ K^{n×r}##
Prove that min##\{rg(A),rg(B)\}≥rg(AB)≥rg(A)+rg(B)−n##
My attempt at a solution
(1) ##AB=(AB_1|...|AB_j|...|AB_r)## (##B_j## is the ##j-th## column of ##B##), I don't know if the following statement is correct: the columns of...
Homework Statement
I have a function f:M_{n×n} \to M_{n×n} / f(X) = X^2.
The questions
Is valid the inverse function theorem for the identity matrix? It talks about the Jacobian at the identity, but I have no idea how get a Jacobian of that function. Can I see the matrices as vectors and...
Hi there!
I'm back again with functions over matrices.
I have a function f : M_{n\times n} \to M_{n\times n} / f(X) = X^2.
Is valid the inverse function theorem for the Id matrix? It talks about the Jacobian at the Id, but I have no idea how get a Jacobian of that function. Can I see that...
1st: Not a specific problem, I just didn't know where else to put it.
We just covered this today in class. Basically what we're doing is reducing higher level matrices to 2x2 matrices and using them to calculate the determinant.
I asked my teacher where that came from, and he was really...
I am going through Mary Boas' "Mathematical Methods in the Physical Sciences 3rd Ed". I finished the chapter 3 section 12 problem set, but I do not understand how she gets eq. 12.39. These don't seem obviously equal to each other. Here is the equation:
$$\lambda Tr=Vr$$ Where T is a matrix...
Is it possible to write series ##\ln x=\sum_na_nx^n##. I am asking you this because this is Taylor series around zero and ##\ln 0## is not defined.
And if ##A## is matrix is it possible to write
##\ln A=\sum_na_nA^n##. Thanks for the answer!
Homework Statement
Find two matrices E and F such that:
EA=
\begin{bmatrix}
2 & 1 & 2\\
0 & 2 & 1\\
0 & 3 & 0\\
\end{bmatrix}
FA=
\begin{bmatrix}
0 & 2 & 1\\
0 & 3 & 0\\
2 & 7 & 2\\
\end{bmatrix}
Homework Equations
The Attempt at a Solution
So I know how to get...
Hi,
i was wondering how the following expression can be decomposed:
Let A=B°C, where B, C are rectangular random matrices and (°) denotes Hadamard product sign. Also, let (.) (.)H denote Hermitian transposition.
Then, AH *A how can be decomposed in terms of B and C ??
For example, AH...
Hi there!
How can I prove that the space of matrices (2x2) nonzero determinant is dense in the space of matrices (2x2) ?
I've already proved that it's an open set.
Thanks.
PD: Sorry about the mistake in the title.
Hi there!
How can I prove that a function which takes an nxn matrix and returns that matrix cubed is a continuous function? Also, how can I analyze if the function is differenciable or not?
About the continuity I took a generic matrix A and considered the matrix A + h, where h is a real...
Homework Statement
For a system Ax= 0, suppose det (A)= .0001. Which of the following describes the solutions to system?
There is exactly one solution, but the system is close to having infinity many.
There is exactly one solution, but the system is close to having none. Homework Equations...
Homework Statement
Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F_13 for i, j = 0,1,2, 3.
Compute F(hat) and verify that F(hat)F = I
Homework Equations
The matrix F(hat) is called the inverse discrete Fourier transform of F.
The Attempt at a Solution
I found that e = 4...
Homework Statement
(i) Verify that 5 is a primitive 4th root of unity in F13.
(ii) Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F13 for i, j = 0,1,2, 3.
Compute F(hat) and verify that F(hat)F= I.
Homework Equations
The matrix F(hat) is called the inverse discrete Fourier...
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...