Matrix Definition and 1000 Threads

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 =...

For this, Dose someone please know how ##ad - bc## and ##-cb + da## are equal to 1? Many thanks!

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...

Let ##M## be a nonzero complex ##n\times n##-matrix. Prove $$\operatorname{rank}M \ge |\operatorname{trace} M|^2/\operatorname{trace}(M^\dagger M)$$ What is a necessary and sufficient condition for equality?

For this, I am not sure what the '2nd and 5th the variables' are. Dose someone please know whether the free variables ##2, 0, 0## from the second column and ##5, 8, \pi##? Or are there only allowed to be one free variable for each column so ##2## and ##5## for the respective columns. Also...

Since Ax = b has no solution, this means rank (A) < m. Since ##A^T y=c## has exactly one solution, this means rank (##A^T##) = m Since rank (A) ##\neq## rank (##A^T##) so matrix A can not exist. Is this valid reasoning? Thanks

My attempt: Let C = $$\begin{pmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{pmatrix}$$ If C is multiplied by B, then: 1) a21 = c21 . b11 0 = c21 . b11 ##\rightarrow c_{21}=0## 2) a31 = c31 . b11 0 = c31 . b11 ##\rightarrow c_{31}=0## 3) a32 =...

My attempt: $$ \begin{vmatrix} 1-\lambda & b\\ b & a-\lambda \end{vmatrix} =0$$ $$(1-\lambda)(a-\lambda)-b^2=0$$ $$a-\lambda-a\lambda+\lambda^2-b^2=0$$ $$\lambda^2+(-1-a)\lambda +a-b^2=0$$ The value of ##\lambda## will be positive if D < 0, so $$(-1-a)^2-4(a-b^2)<0$$ $$1+2a+a^2-4a+4b^2<0$$...

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...

This question is related to equation (1),(3), and (4) in the [paper][1] [1]: https://arxiv.org/abs/2002.12252

How do I calculate a 3x3 matrix multiplication with a 3 column row vector, like below? ## \begin{bmatrix} A11 & A12 & A13\\ A21 & A22 & A23\\ A31 & A32 & A33 \end{bmatrix}\begin{bmatrix} B1 & B2 & B3 \end{bmatrix} ##

The attempt at a solution: I tried the normal method to find the determinant equal to 2j. I ended up with: 2j = -4yj -2xj -2j -x +y then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options: 1) 0= y(-4j-j^2) -x(2j-1) -2j 2)...

It is easy to see that a matrix of the given form is actually an unitary matrix i,e, satisfying AA^*=I with determinant 1. But, how to see that an unitary matrix can be represented in the given way?

Let ## \mathbf{x''} = A\mathbf{x} ## be a homogenous second order system of linear differential equations where ## A = \begin{bmatrix} a & b\\ c & d \end{bmatrix} ## and ## \mathbf{x} = \begin{bmatrix} x(t)\\ y(t)) \end{bmatrix} ## Now to solve this equation we transform it into a 4x4...

How to derive (proof) the following trace(A*Diag(B*B^T)*A^T) = norm(W,2), where W = vec(sqrt(diag(A^T*A))*B) & sqrt(diag(A^T*A)) is the square root of diag(A^T*A), B & A are matrix. Please see the equation 70 and 71 on page 2068 of the supporting matrial.

Hi all, I want to know if a second solution exists for the following math equation: Ce^{At} ρ_p+(CA)^{−1} (e^{At}−I)B=0 Where C, ρ_p, A and B are constant matrices, 't' is scalar variable. I know that atleast one solution i.e. 〖t=θ〗_1 exists, but I want a method to determine if there is...

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...

TL;DR Summary: For every Complex matrix proove that: (Y^*) * X = complex conjugate of {(X^*) * Y} Here (Y^*) and (X^*) is equal to complex conjugate of (Y^T) and complex conjugate of (X^T) where T presents transponse of matrix I think we need to use (A*B)^T= (B^T) * (A^T) and Can you help...

Until now in my studies - matrices were indexed like ##M_{ij}##, where ##i## represents row number and ##j## is the column number. But now I'm studying vectors, dual vectors, contra- and co-variance, change of basis matrices, tensors, etc. - and things are a bit trickier. Let's say I choose to...

I'm trying to find the purification of this density matrix $$\rho=\cos^2\theta \ket{0}\bra{0} + \frac{\sin^2\theta}{2} \left(\ket{1}\bra{1} + \ket{2}\bra{2} \right) $$ So I think the state (the purification) we're looking for is such Psi that $$ \ket{\Psi}\bra{\Psi}=\rho $$ But I'm not...

The mixing of the 3 generations of fermions are tabulated into the CKM matrix for quarks: $$ \begin{bmatrix} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\sigma_{13}} \\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\sigma_{12}} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\sigma_{13}} & s_{23}c_{13} \\...

If we have an arbitrary square matrix populated randomly with 1s and 0s, is there an operator which will return a unique number for each configuration of 1s and 0s in the matrix? i.e. an operation on $$ \begin{pmatrix} 1 &0 &0 \\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix} $$ would return something...

## A = \pmatrix{ -4 & -3 & 3 & 3 \\ -3 & -4 & 3 & 3 \\ -6 & -3 & 5 & 3 \\ -3 & -6 & 3 & 5 } ## over ## \mathbb{R}##. Let ## T_A: \mathbb{R}^4 \to \mathbb{R}^4 ## be defined as ## T_A v = Av ##. Thus, ## T_A ## represents ## A ## in the standard basis, meaning ## [ T_A]_{e} = A ##. I've...

From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix: $$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b...

I know that if ##\eta_{\alpha'\beta'}=\Lambda^\mu_{\alpha'} \Lambda^\nu_{\beta'} \eta_{\alpha\beta}## then the matrix equation is $$ (\eta) = (\Lambda)^T\eta\Lambda $$ I have painstakingly verified that this is indeed true, but I am not sure why, and what the rules are (e.g. the ##(\eta)## is in...

$$\begin{bmatrix} 4& 3 \\ 6 & -2 & \\ \end{bmatrix}= \begin{bmatrix} 1& 0 \\ a& 1& \\ \end{bmatrix}⋅ \begin{bmatrix} b& c \\ 0& d& \\ \end{bmatrix}$$ ##b=4, ab=6,⇒b=1.5, d=-6.5, c=3## $$\begin{bmatrix} 4& 3 \\ 6 & -2 & \\ \end{bmatrix} = \begin{bmatrix} 1& 0\\ \dfrac{3}{2} & 1 & \\...

I have learnt about the power iteration for any matrix say A. How it works is that we start with a random compatible vector v0. We define vn+1 as vn+1=( Avn)/|max(Avn)| After an arbitrary large number of iterations vn will slowly converge to the eigenvector associated with the dominant...

Hi, For a 2 x 2 matrix ##A## representing a Markov transitional probability, we can compute the stationary vector ##x## from the relation $$Ax=x$$ But can we compute ##A## of the 2x2 matrix if we know the stationary vector ##x##? The matrix has 4 unknowns we should have 4 equations; so for a ##A...

David Tong gives an interesting talk about the lattice chiral fermion problem here. https://weblectures.leidenuniv.nl/Mediasite/Channel/ehrenfestcolloquium/watch/5de33fbc14cd4595a6614ca7683bf71e1d Abstract: Are we living in the matrix? No. Obviously not. It's a daft question. But, buried...

T(α1), T(α2), T(α3) written in terms of β1, β2: Tα1 =(1,−3) Tα2 =(2,1) Tα3 =(1,0). Then there is row reduction: Therefore, the matrix of T relative to the pair B, B' is I don't understand why the row reduction takes place? Also, how do these steps relate to ## B = S^{-1}AS ##? Thank you.

I would appreciate help walking through this. I put solid effort into it, but there's these road blocks and questions that I can't seem to get past. This is homework I've assigned myself because these are nagging questions that are bothering me that I can't figure out. I'm studying purely on my...

Hi, I have a 2x2 hermitian matrix like: $$ A = \begin{bmatrix} a && b \\ -b && -a \end{bmatrix} $$ (b is imaginary to ensure that it is hermitian). I would like to find an orthogonal transformation M that makes A skew-symmetric: $$ \hat A = \begin{bmatrix} 0 && c \\ -c && 0 \end{bmatrix} $$ Is...

Part (A): The matrix is a singular matrix because the determinant is 0 with my calculator. Part (B): Once I perform Gauss Elimination with my pivot being 0.6 I arrive at the last row of matrix entries which are just 0's. So would this be why Gauss Elimination for partial pivoting fails for this...

This screenshot contains the original assignment statement and I need help to solve it. I have also attached my attempt below. I need to know if my matrices were correct and my method and algebra to solve the problem was correct...

I have found J^2 and Jz, but I am not sure how to find Jx and Jy. I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results. Thanks!

Hi, I have been studying the Fisher matrix to apply in a project. I understand how to compute a fisher matrix when you have a simple model for example which is linear in the model parameters (in that case the derivatives of the model with respect to the parameters are independent of the...

If ##U## is an unitary operator written as the bra ket of two complete basis vectors :##U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|## ##U^\dagger=\sum_{k}\left|a^{(k)}\right\rangle\left\langle b^{(k)}\right|## And we've a general vector ##|\alpha\rangle## such that...

Hello, I am often designing math exams for students of engineering. What I ask is the following: Can I choose any real 3x3 symmetric matrix with positive eigenvalues as a realistic matrix of inertia? Possibly, there are secret connections between the off-diagonal elements (if not zero)...

Let's assume that ##A## is unitary and diagonalisable, so, we have ## \Lambda = C^{-1} A C ## Since, ##\Lambda## is made up of eigenvalues of ##A##, which is unitary, we have ## \Lambda \Lambda^* = \Lambda \bar{\Lambda} = I##. I tried using some, petty, algebra to prove that ##C C* = I## but...

I came across a statement in《A First Course in General Relativity》:“The only matrix diagonal in all frames is a multiple of the identity：all its diagonal terms are equal.”Why?I don’t remember this conclusion in linear algebra.The preceding part of this sentence is:Viscosity is a force parallel...

I have a variance-covariance matrix W with diagonal elements diag(W). I have a vector of weights v. I want to scale W with these weights but only to change the variances and not the covariances. One way would be to make v into a diagonal matrix and (say V) and obtain VW or WV, which changes both...

Hello! I have this system here $$ \left[ \begin{matrix} -2 & 4 & \\\ 1 & -2 & {} \end{matrix} \right]x +\begin{pmatrix} 2 \\\ y \end{pmatrix}u $$ Now although the problem is for my control theory class,the background is completely math(as is 90% of control theory) Basically what I need to...

Does anyone know a C# class that can return a value (0 - 100 percentage) of How close a perfect gaussian curve an 2D Matrix is? for example, these would all return a 100%:

I’m really unable to solve those questions which ask to find a nonsingular ##C## such that $$ C^{-1} A C$$ is a digonal matrix. Some people solve it by finding the eigenvalues and then using it to form a diagonal matrix and setting it equal to $$C^{-1} A C$$. Can you please tell me from scratch...

Summary: The transition rate matrix for a problem where there are 5 Processing Units A computer has five processing units (PU’s). The lifetimes of the PU’s are independent and have the Exp(µ) law. When a PU fails, the computer tries to reconfigure itself to work with the remaining PU’s. This...

Summary: Finding the transition matrix of a paint ball game where only 3 probabilities are given. We have the following question: Alice, Tom, and Chloe are competing in paint ball. Alice hits her target 40% of the time, Tom hits his target 25% of the time, and Chloe hits her target 30% of the...

Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation. I suppose I could write the...

This is related to a recent (mainly unserious) post I recently made. I did some more work on a similar problem and I'd like to bounce off an idea why this doesn't work. I really am not sure if I'm right so I'd appreciate a comment. I am working with some simple systems of difference...

I'm looking at ways of solving 2nd order difference equations with non-constant coefficients. I am working on a method to use transformations (ie rewriting the equation in new variables) to change the form of the equation. Such as a_{n + 2} + f(n) a_{n +1} + g(n) a_n = 0 to something like u_{n...

In geometry, a vector ##\vec{X}## in n-dimensions is something like this $$ \vec{X} = \left( x_1, x_2, \cdots, x_n\right)$$ And it follows its own laws of arithmetic. In Linear Analysis, a polynomial ##p(x) = \sum_{I=1}^{n}a_n x^n ##, is a vector, along with all other mathematical objects of...