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.
Some reflections in the plane can be represented by a rotation in three dimensions, and some cannot: e.g., reflections across the x or y axes can. but a 2D reflection across the line x=y cannot. Thus the question in the summary.
Hi all,
I was wondering if there was a clean/closed form version of the following expression: $$e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}$$
where ##X,Y,Z## are matrices that don't commute with each other. I know of the BCH identity ##e^{X}Ye^{-X} = Y + [X,Y] + \frac{1}{2!}[X,[X,Y]] +...
Hi Pfs,
I read a paper about the Cabibbo matrix and the CKM matrix.
The first one is a 2*2 real matrix and the other a 3*3 matrix with complex entries.
In this article i read that a n*n matrix has 2 n*n real degrees or freedom.
The unitarity (orthonormal basis) devides this number by 2.
I read...
Hi,
In my linear algebra homework, there is a bonus assignment where we are supposed to use Mathematica to calculate matrices and their determinants etc. here is the assignment.
Unfortunately, I am a complete newbie when it comes to Mathematica, this is the first time I have worked with...
Since ##AB = B##, so matrix ##A## is an identity matrix.
Similarly, since ##BA = A## so matrix ##B## is an identity matrix.
Also, we can say that ##A^2 = AA=IA= A## and ##B^2 = BB=IB= B##.
Therefore, ##A^2 + B^2 = A + B## which means (a) is a correct answer.
Also we can say that ##A^2 + B^2 =...
I feel if we have the matrix equation X = AB, where X,A and B are matrices of the same order, then if we apply an elementary row operation to X on LHS, then we must apply the same elementary row operation to the matrix C = AB on the RHS and this makes sense to me. But the book says, that we...
Hi! Please, could you help me on how to solve the following matrix ?
I need to replace the value 3 on the third line by 0, the first column need to remain zero and 1 for the third column. I'm having a lot of difficulties with this. How would you proceed ?
Thank you for your time and help...
In a permutation matrix (the identity matrix with rows possibly rearranged), it is easy to spot those rows which will indicate a fixed point -- the one on the diagonal -- and to spot the pairs of rows that will indicate a transposition: a pair of ones on a backward diagonal, i.e., where the...
One way would be to assume
$$A= \begin{bmatrix}a_1 & a_2\\a_3 & a_4 \end{bmatrix}$$ and $$B=\begin{bmatrix}b_1 & b_2\\b_3 & b_4\end{bmatrix}$$ and then multiply but then you end up with 4 equations and 8 variables, how would that work?
the other way would be to use trial and error, any help...
As there was quite rightly some criticism earlier about not following proper theory, I will first demonstrate what I have understood of the gamma matrices of SU(3).
There are 8 gamma matrices that together generate the SU(3) group used in QCD. Gell-Mann used only 2, ##\gamma_3## and...
I learned that for a bilinear form/square form the following theorem holds:
matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form.
Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing...
I have a different way in solving the problem, but strangely, the result is different from that written in the solution manual.
My method:
Firstly, we will solve the ##AB=A## equation
$$AB=A$$
$$B=A^{−1}A$$
$$B=I$$
where ## I## is an identity matrix
Similarly, we can solve ##BA=B## using the...
We are given f(x)=(1/2)(xT)Qx+qTx-B where xk+1=xk+αksk, the search direction is sk=-∇f(xk). Q is a 2x2 matrix and q is 2x1 matrix and B=6. My issue is I'm confused what -∇f(xk) is, is ∇f(xk)=Q(xk)-q? Just like how it is in Conjugate Gradient/Fletcher Reeve's method? Or is it Q(xk)+q?
Thank you
Square matrices are closed under addition and their own form of multiplication, but in general do not commute.
What algebraic structure then describes this, along with polynomials of matrices and allows us to amend with other operations, such as differentiation or integration defined on these...
I'm glad there's a section here dedicated to differential equations.
I've seen in the fundamental theorem of linear ordinary systems, that, for a real matrix ##A##, we have ## d/dt \exp(At) = A \exp(At)##. I'm wondering if there are analogs of this, like for instance, generalizing a system of...
As I was looking for an example for a metric tensor that isn't among the usual suspects, I observed that the Cartan matrix I wanted to use is positive definite (I assume all are), but not symmetric. Are the symmetry breaks in quantum physics related to this fact?
We mainly have to prove that this quantity
## \bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi} ##
is greater or equal than zero for all ##\ket{\varphi}##.
Being ##\ket{\varphi}## a product state it is straightforward to demonstrate such inequality. I...
1. A is a matrix of order 2x2 whose main diagonal's entries' sum is zero. Prove that A^2 is a scalar matrix.
2. Given: A and B are two matrices of order 2x2. Prove that the sum of the entries of the main diagonal of AB-BA is zero.
3. A, B and C are three matrices of order 2x2. Given: A^2 is a...
Going through Axler's awful book on linear algebra. The complex spectral theorem (for operator T on vector space V) states that the following are equivalent: 1) T is normal 2) V has an orthonormal basis consisting of eigenvectors of T and 3) the matrix representation of T is diagonal with...
I'm really unable to have a start, because I cannot think of any matrix (other than ##O##) such that its cube is the zero matrix. I tried to assume A = ##\begin{bmatrix} a &c \\b &d \end{bmatrix} ## and computed ##A^3## and set it to ##O## to get an idea how the elements would look like, but the...
When a matrix is represented as a box it seems all very clear, but this representation
$$
A = (a_{ij} )_{i, j =1}^{m,n}$$
Isn't very representative to me. The i -j thing creates a lot of confusion, when we write ##a_{ij}## do we mean the element of i th row and jth column or the other way...
I am working with matrices to balance chem equations. I have googled hard equations to balance and find that most, at least what I have found, have the number of elements to be less than the terms in the equation. for example
... FeCr2O4 + ... Na2CO3 + ... O2 -->... Na2CrO4 + ...Fe2O3 + 1CO...
Wolfgang Pauli's matrices are
$$\sigma_x=\begin{bmatrix}0& 1\\1 & 0\end{bmatrix},\quad \sigma_y=\begin{bmatrix}0& -i\\i & 0\end{bmatrix},\quad \sigma_z=\begin{bmatrix}1& 0\\0 & -1\end{bmatrix}$$
He introduces these equations as "the equations of motion" of the spin in a magnetic field.
$$...
Dear Everybody,
I have some trouble with this problem: Finding a sequence of elementary matrix for this matrix A.
Let ##A=\begin{bmatrix} 4 & -1 \\ 3& -1\end{bmatrix}##. I first used the ##\frac{1}{4}R1##-> ##R1##. So the ##E_1=\begin{bmatrix} \frac{1}{4} & 0 \\ 0& 1\end{bmatrix}##. So the...
Hi Pfs , happy new year.
I wonder if there is a problem with the manner i see density matrices:
I use to consider them without a statistical point of view , just like i do with Hilbert vectors. no more no less. So the points on the Block sphere are only pecular points of those which are inside...
What is the basis of 2x2 matrices with real entries? I know that the basis of 2x2 matrices with complex entries are 3 Pauli matrices and unit matrix:
\begin{bmatrix}
0 & 1 \\[0.3em]
1 & 0 \\[0.3em]
\end{bmatrix},
\begin{bmatrix}
0 & -i \\[0.3em]
i & 0...
Hi,
I'd like to have a little insight about why the determinants of ℝ2x2 and ℝ3x3 matrices are computed that way.
I know how to calculate said determinants in both the cases and I also know what's the meaning behind it thanks to "3blue1brown"'s youtube channel, which states that they are a...
If I understand this correctly, this is the right answer: ##M \begin{pmatrix} 0.2\\ 0.1\end{pmatrix}##
There is an inverse matrix in the next question:
Continuing with the previous problem, let ##\vec v = M^{-1} \begin{pmatrix} 0.2\\ 0.1\end{pmatrix}##, where ##M^{-1}## is the inverse matrix of...
Let ##A## be a matrix of size ##(n,n)##. Denote the entry in the i-th row and the j-th column of ##A## by ##a_{ij}##, for some ##i,j\in\mathbb{N}##. For brevity, we call ##a_{ij}## entry ##(i,j)## of ##A##.
Define the matrix ##X## to be of size ##(n,n)##, and denote entry ##(i,j)## of ##X## as...
Hi
Several attention mechanisms require trainable matrices and vectors. I have been trying to learn how to implement this in Tensorflow w/ Keras. Every implementation I see use the Dense layer from Keras, but I have a tendency to get lost trying to understand why and what they do afterwards...
Is the following a correct demonstration that quantum mechanics can be done in a real vector space?
If you simply stack the entries of density matrices into a column vector, then the expression ##\textrm{Tr}(AB^\dagger)## is the same as the dot product in a complex vector space (Frobenius inner...
Hi guys! :)
I was solving some linear algebra true/false (i.e. prove the statement or provide a counterexample) questions and got stuck in the following
a) There is no ##A \in \Bbb R^{3 \times 3}## such that ##A^2 = -\Bbb I_3## (typo corrected)
I think this one is true, as there is no squared...
Hi,
I haven't made a code for it yet, however I will be making a Gauss-Seidel algroithm in Fortran 90, for solving very large matrices (actually to initiate the multi-gird method but that's irrelevant for this). As part of this, I wish to insert matrices into other matrices.
in MATLAB this...
i am new to MATLAB and and as shown below I have a second order differential equation M*u''+K*u=F(t) where M is the mass matrix and K is the stifness matrix and u is the displacement.
and i have to write a code for MATLAB using ODE45 to get a solution for u. there was not so much information on...
What branch of mathematics studies multinomial functions of matrices? ( i.e matrix valued functions of square matrices such as ##f(A,B,C) = ABC + BAC + 2A^2 + 3C##)
Hey! :giggle:
The set of $2$-dimensional orthogonal matrices is given by $$O(2, \mathbb{R})=\{a\in \mathbb{R}^{2\times 2}\mid a^ta=u_2\}$$ Show the following:
(a) $O(2, \mathbb{R})=D\cup S$ and $D\cap S=\emptyset$. It holds that $D=\{d_{\alpha}\mid \alpha\in \mathbb{R}\}$ and...
Hello there I am having trouble with part b) of this exercise. I can apply the rotation matrix easily enough and get:
$$
R(-\theta) \vec J= \begin{bmatrix} A\cos\theta + B\sin{\theta}e^{i\delta} \\
-A\sin\theta + B\cos{\theta}e^{i\delta} \end{bmatrix}
$$
I decided to convert the exponential...
I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes).
Now I would like to make the cross-correlations synthesis of these 2 matrixes by applying for each parameter the well known formula (coming from...
$\tiny{311.1.1.26}$
Construct 3 augmented matrices for linear systems whose solution set is $x_1=3, \quad x_2=-2, \quad x_3=-1$
ok the only thing I could think of is just rearrange the rows of an RREF matrix. albeit losing the triangle format
hopefully no typos
$\left[\begin{array}{rrr|rr}
1&...
Summary:: Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix and let ##\lambda \in \Bbb R## be an eigenvalue of ##A##. Prove that the geometric multiplicity ##g(\lambda)## of ##A## equals its algebraic multiplicity ##a(\lambda)##.
Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix...
Hello all,
I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the
row space to the null space
column space to the left null space
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In the book of G. Strang there is this nice picture...