Matrix Definition and 1000 Threads

Homework Statement Being f : ℝ4 → ℝ4 the endomorphism defined by: ƒ((x, y, z, t)) = (3x + 10z, 2y - 6z - 2t, 0, -y+3z+t) Determine the base and dimension of Im(ƒ) and Ker(ƒ). Complete the base you chose in Im(ƒ) into a base of R4. Homework Equations Matrix A: $$\begin {bmatrix} 3 & 0 & 10 &...

I'm doing an online course in quantum information theory, but it seems to require some knowledge of linear algebra that I don't have. A definition that popped up today was the definition of the absolute value of a matrix as: lAl = √(A*A) , where * denotes conjugate transpose. Now for a...

Can an orthogonal matrix involve complex/imaginary values?

Homework Statement Determine the values of h such that the matrix is the augmented matrix of a consistent linear system. 1 4 -2 3 h -6 The attempt at a solution The answer I got differs from the back of the book. I tried solving it by adding R1(4) to R2 1 3 -2 -4 h 8 becomes 1...

Hi, my high school students enjoy using the applet found here (http://pages.jh.edu/~virtlab/bridge/truss.htm) to design model (basswood) bridges for our annual regional contest. It seems to require firefox these days. Recently, some designs have been causing extremely large forces to be...

Homework Statement [/B] 1. I've been tasked with forming a 10 x 10 matrix with elements 0, 1, 2, 3, 4, 5,... and have it display properly. 2. Then, take this matrix and make a 2d-histogram out of it. Homework Equations Here is my code void matrix6( const int n = 10) { float I[n][n]; //...

Homework Statement Consider the LTI (A,B,C,D) system $$ \dot{x}= \begin{pmatrix} 0.5&0&0&0\\ 0&-2&0&0\\ 1&0&0.5&0\\ 0&0&0&-1 \end{pmatrix} x+ \begin{pmatrix} 1\\ 1\\ 0\\ 0 \end{pmatrix} u $$ $$ y= \begin{pmatrix} 0&1&0&1 \end{pmatrix} x $$ Determine if A is diagonalizable Homework EquationsThe...

Hello all, I have a matrix A and I am looking for it's eigenvalues. No matter what I do, I find that the eigenvalues are 0, 1 and (k+1), while the answer of both the book and Maple is 0 and (k+2). I tried two different technical approaches, both led to the same place. The matrix is...

Although strictly quantum mechanics is defined in ##L_2## (square integrable function space)， non normalizable states exists in literature. In this case, textbooks adopt an alternative normalization condition. for example, for ##\psi_p(x)=\frac{1}{2\pi\hbar}e^{ipx/\hbar}## ##...

I have a doubt... Look this matrix equation: \begin{bmatrix} A\\ B \end{bmatrix} = \begin{bmatrix} +\frac{1}{\sqrt{2}} & +\frac{1}{\sqrt{2}}\\ +\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} X\\ Y \end{bmatrix} \begin{bmatrix} X\\ Y \end{bmatrix} = \begin{bmatrix}...

Due to the definition of spin-up (in my project ), \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} as opposed to \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} and the annihilation operator is...

Homework Statement X= 1st row: (0, 1, 0, 0), 2nd row: (1, 0, 0, 0), 3rd row: (0, 0, 0, 1-i), 4th row: (0, 0, 1+i, 0) Find the eigenvalues and eigenvectors of the matrix X. Homework Equations |X-λI|=0 (characteristic equation) (λ is the eigenvalues, I is the identity matrix) (X-λI)V=0 (V is the...

There is something that I don't quite understand or want clarification. See John Wheeler article "100 years of the quantum" http://arxiv.org/pdf/quant-ph/0101077v1.pdf refer to page 6 with parts of the quotes read "so if we could measure whether the card was in the alpha or beta-states, we...

Homework Statement Let A(l) = [ 1 1 1 ] [ 1 -1 2] be the matrix associated to a linear transformation l:R3 to R2 with respect to the standard basis of R3 and R2. Find the matrix associated to the given transformation with respect to hte bases B,C, where B = {(1,0,0) (0,1,0) , (0,1,1) } C =...

If we have two square matrices of the same size P and Q, we can put one in the exponent of the other by: M = P^Q = e^{ln(P)Q} ln(P) may give multiple results R, which are square matrices the same size as P. So then we have: M = e^{RQ} which can be Taylor expanded to arrive at a final square...

Hello! (Wave) I want to solve the following linear programming problem: $$\min (5y_1-10y_2+7y_3-3y_4) \\ y_1+y_2+7y_3+2y_4=3 \\ -2y_1-y_2+3y_3+3y_4=2 \\ 2y_1+2y_2+8y_3+y_4=4 \\ y_i \geq 0, i \in \{ 1, \dots, 4 \}$$ $\begin{bmatrix} 1 & 1 & 7 & 2 & | & 3\\ -2 & -1 & 3 & 3 & | & 2\\ 2 & 2 & 8...

Let the operators ##\hat{A}## and ##\hat{B}## be ##-i\hbar\frac{\partial}{\partial x}## and ##x## respectively. Representing these linear operators by matrices, and a wave function ##\Psi(x)## by a column vector u, by the associativity of matrix multiplication, we have...

Happy new year. Why everybody uses this definition of rotation matrixR(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\[0.3em] \sin\theta & \cos\theta \\[0.3em] \end{bmatrix} ? This is clockwise rotation. And we always use counter clockwise in...

Why isn't the second line in (5.185) ##\sum_k\sum_l<\phi_m\,|\,A\,|\,\psi_k><\psi_k\,|\,\psi_l><\psi_l\,|\,\phi_n>##? My steps are as follows: ##<\phi_m\,|\,A\,|\,\phi_n>## ##=\int\phi_m^*(r)\,A\,\phi_n(r)\,dr## ##=\int\phi_m^*(r)\,A\,\int\delta(r-r')\phi_n(r')\,dr'dr## By the closure...

Each set of constant numbers such as ##(v_1, v_2, v_3)## are the components of a constant Cartesian vector because by rotation of coordinates they satisfy the transformation rule. Can we consider each set of constant arrays ## a_{ij};i,j=1,2,3 ## as components of a Cartesian tensor? In other...

For a state |\Psi(t)\rangle = \sum_{k}c_k e^{-iE_kt/\hbar}|E_k\rangle , the density matrix elements in the energy basis are \rho_{ab}(t) = c_a c^*_be^{-it(E_a -E_b)/\hbar} How is it that in the long time limit, this reduces to \rho_{ab}(t) \approx |c_a|^2 \delta_{ab} ? Is there some...

How do you write a matrix such as below image in Latex, in this forum?

Hello! (Wave) We have that $q(x) \geq q_0>0, x \in [0,1], h>0$. Suppose that we have this $(N+1) \times (N+1) matrix$: $\begin{bmatrix} \frac{1}{h^2}+\frac{1}{h}+\frac{q(x_0)}{2} & -\frac{1}{h^2} & 0 & \cdots & 0 \\ -\frac{1}{h^2} & \frac{2}{h^2}+q(x_1) & -\frac{1}{h^2} & \cdots &0 \\ 0 &...

Hey all. I know that A^TA is positive semidefinite. Is it possible to achieve a positive definite matrix from such a matrix multiplication (taking into account that A is NOT necessarily a square matrix)?

Homework Statement Determine if the matrix is diagonalizable or not. A= [ 3 -1 ] [ 1 1 ] Homework Equations Eigenvalues = det(A-Iλ) determinant of a 2x2 matrix = ad-bc The Attempt at a Solution Eigenvalues = det(A-Iλ) [ 3 -1 ] - [ λ 0 ] = [ 3 -λ -1 ] [ 1 1 ] [ 0 λ ]...

Fredrik submitted a new PF Insights post Matrix Representations of Linear Transformations Continue reading the Original PF Insights Post.

Homework Statement Hi this isn't really a question but more so understanding an example that was given to me that I not know how it came to it's conclusion. This is a question pertaining linear transformation for coordinate isomorphism between basis. https://imgur.com/a/UwuACHomework Equations...

Hello hope you can help me. Can anybody tell me what goes on from equation 3 to 4. especially how gets in?

Homework Statement Let A be an nxn matrix, and let |v>, |w> ∈ℂ. Prove that (A|v>)*|w> = |v>*(A†|w>) † = hermitian conjugate Homework EquationsThe Attempt at a Solution Struggling to start this one. I'm sure this one is likely relatively quick and painless, but I need to identify the trick...

Homework Statement Diagonalize matrix using only row/column switching; multiplying row/column by a scalar; adding a row/column, multiplied by some polynomial, to another row/column. Homework EquationsThe Attempt at a Solution After diagonalization I get a diagonal matrix that looks like...

Hi, Concerning optical polarization, what is the Jones Matrix of a mirror at a non-zero angle of incidence with respect to incoming light? For a mirror at normal incidence the matrix is (1 0; 0 -1); How do I incorporate the angle?

Homework Statement A = \begin{bmatrix} 2 & 1 & 0\\ 0& -2 & 1\\ 0 & 0 & 1 \end{bmatrix} Homework EquationsThe Attempt at a Solution The spectrum of A is \sigma (A) = { \lambda _1, \lambda _2, \lambda _3 } = {2, -2, 1 } I was able to calculate vectors v_1 and v_3 correctly out of the...

Homework Statement How many hadamard matrices exists for size n? Homework Equations Hadamard matrices are square matrices whose entries are either +1 or −1 and whose rows are mutually orthogonal. The Attempt at a Solution I am just curious how many exists for 4, 8 and in general.[/B]

Hi, I was wondering if it's possible to colour the rows and columns of a matrix in mathematica. I have received help from another forum and the code of my matrix is the following: Rasterize@ Style[MatrixForm[{{n, -1 + n, -2 + n, \[CenterEllipsis], 1}, {2 n, 2 n - 1, 2 n - 2...

I find that the quark mixing factor say for example ##V_{ub}## is the same for: u ##\Leftrightarrow## b ##u\Leftrightarrow\bar{b}## ##\bar{u}\Leftrightarrow## b ##\bar{u}\Leftrightarrow\bar{b}## Does this have something to do with weak interaction being unable to distinguish these from one...

An exercise in my text requires me to (in MATLAB) generate a numeric solution to a given second order differential equation in three different ways using a forwards, centered and backwards difference matrix. I got reasonable answers for \vec{u} that agreed with each other (approximately) for the...

The question mentions an orthogonal matrix describing a rotation in 3D ... where $\phi$ is the net angle of rotation about a fixed single axis. I know of the 3 Euler rotations, is this one of them, arbitrary, or is there a general 3-D rotation matrix in one angle? If I build one, I would start...

Maybe I just need help understanding the question ... write $ x^2 + 2xy + 2yz + z^2 $ as a sum of squares $ (x')^2 -2(y')^2 + 2(z')^2 $ in a rotated coord system. The 1st expression $ = \left[ x, y, z \right]M \begin{bmatrix}x\\y\\z\end{bmatrix} $ and I get $ M =...

Starting with the orbital angular momentum of the ith element of mass, $ \vec{L}_I = \vec{r}_I \times \vec{p}_I = m_i \vec{r}_i \times \left( \omega \times \vec{r}_i\right) $, derive the inertia matrix such that $\vec{L} =I\omega, |\vec{L} \rangle = I |\vec{\omega} \rangle $ I used a X b X c...

Show that the eigenvalues of any matrix are unaltered by a similarity transform - the book says this follows from the invariance of the secular equation under a similarity transform - which is news to me. The secular eqtn is found by $$Det(A-\lambda I)=0$$ and is a poly in $$\lambda $$, so I...

Homework Statement I have to make program that a user inputs a matrix and program displays it.Homework EquationsThe Attempt at a Solution I know the logic as in c++ I am able to display that. Here, m=input('Enter rows of matrix'); % Why not double quotes here as in cout of C++? n=input('Enter...

Homework Statement My Program is not showing the sum value or not returning it. A blank space is coming.Why that is so? Homework Equations Showing the attempt below in form of code. The Attempt at a Solution #include<iostream.h> #include<conio.h> Prime_Sum(int arr[30][30],int m, int n); void...

I have an exercise which says to show that for vectors, $ A \cdot A^{-1} = A^{-1} \cdot A = I $ does NOT define $ A^{-1}$ uniquely. But, let's assume there are at least 2 of $ A^{-1} = B, C$ Then $ A \cdot B = I = A \cdot C , \therefore BAB = BAC, \therefore B=C$, therefore $ A^{-1}$ is...

I'm not sure I have the right approach here: Using the three 2 X 2 Pauli spin matrices, let $ \vec{\sigma} = \hat{x} \sigma_1 + \hat{y} \sigma_2 +\hat{z} \sigma_3 $ and $\vec{a}, \vec{b}$ are ordinary vectors, Show that $ \left( \vec{\sigma} \cdot \vec{a} \right) \left( \vec{\sigma} \cdot...

Given a Positive Definite Matrix ## A \in {\mathbb{R}}^{2 \times 2} ## given by: $$ A = \begin{bmatrix} {A}_{11} & {A}_{12} \\ {A}_{12} & {A}_{22} \end{bmatrix} $$ And a Matrix ## B ## Given by: $$ B = \begin{bmatrix} \frac{1}{\sqrt{{A}_{11}}} & 0 \\ 0 & \frac{1}{\sqrt{{A}_{22}}}...

Hello everyone, I have a question that will probably turn out to be trivial. I have the following matrix: $$ U=\text{diag}(e^{2i\alpha},e^{-i\alpha},e^{-i\alpha}). $$ This seems to me as an SU(2) matrix in the adjoint representation since it's unitary and has determinant 1. Am I right? If so...

Homework Statement Find the eigenvalues of the matrix ## \left( \begin{array}{cc} 3 & -1.5\\ -1.5 & -1\\ \end{array} \right) ## It's probably a really stupid mistake, but the answer I get doesn't match the answer from wolfram alpha's eigenvalue calculator... always a bad sign. Homework...

Show that if A and B are square matrices of the same size such that B is an invertible matrix, then A must be a zero matrix.

So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...

I'm trying to show that A be a 3 x 3 upper triangular matrix with non-zero determinant . Show by explicit computation that A^{-1}(inverse of A) is also upper triangular. Simple showing is enough for me. \begin{bmatrix}\color{blue}a & \color{blue}b & \color{blue}c \\0 & \color{blue}d &...