What is Matrix: Definition and 1000 Discussions

The Multistate Anti-Terrorism Information Exchange Program, also known by the acronym MATRIX, was a U.S. federally funded data mining system originally developed for the Florida Department of Law Enforcement described as a tool to identify terrorist subjects.
The system was reported to analyze government and commercial databases to find associations between suspects or to discover locations of or completely new "suspects". The database and technologies used in the system were housed by Seisint, a Florida-based company since acquired by Lexis Nexis.
The Matrix program was shut down in June 2005 after federal funding was cut in the wake of public concerns over privacy and state surveillance.

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  1. R

    I The Complete Solution to the matrix equation Ax = b

    We are trying to find the complete solution to the matrix equation ##A\vec x = \vec b## where A is an m x n matrix and ##\vec b## can be anything except the zero vector. The entire solution is said to be: ##\vec x = \vec x_p + \vec x_n## where ##\vec x_p## is the solution for a particular ##\vec...
  2. S

    I Is there any matrix equivalent for the Clifford product?

    Well, the question is in the title.
  3. S

    I What is the proper matrix product?

    It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations: 1) matrix addition; 2) multiplication by the undelying field elements; 3) matrix multiplication. Is the last one...
  4. S

    Setting Free variables when finding eigenvectors

    upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations. So I took b as a free variable to solve the equation int he following way. But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free...
  5. Leonardo Machado

    I Chebyshev Differentiation Matrix

    Hi everyone. I am studying Chebyshev Polynomials to solve some differential equations. I found in the literature that if you have a function being expanded in Chebyshev polynomials such as $$ u(x)=\sum_n a_n T_n(x), $$ then you can also expand its derivatives as $$ \frac{d^q u}{dx^q}=\sum_n...
  6. S

    Matrix concept Questions (invertibility, det, linear dependence, span)

    I have a trouble showing proofs for matrix problems. I would like to know how A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix holds for square matrix A. It would be great if you can show how one leads to another with examples! :) Thanks for helping...
  7. LCSphysicist

    Spring constant matrix and normal modes (4 springs and 3 masses)

    We need to find the normal modes of this system: Well, this system is a little easy to deal when we put it in a system and solve the system... That's not what i want to do, i want to try my direct matrix methods. We have springs with stiffness k1,k2,k3,k4 respectively, and block mass m1, m2...
  8. B

    Trouble with Matrix Exponentials

    I've attempted to solve this by separating A into a diagonal matrix D and nilpotent matrix N: D = {{1, 0}, {0, 0}} N = {{0, 0}, {1, 0}} e^(At) = e^((D + N)t) = e^(Dt) * e^(Nt) When N is raised to the second power, it becomes the zero matrix. Therefore, e^(Nt) = I + Nt = {{1, 0}, {t, 1}} Note...
  9. M

    MATLAB Padding cells to make a single matrix

    Hi PF! I'd like to make one matrix from a cell. I've checked several suggestions, the most promising here but this did not work, giving me the unhelpful error Brace indexing is not supported for variables of this type. Error in cell2mat (line 42) cellclass = class(c{1}); Error in Feven (line...
  10. S

    Using a determinant to find the area of the triangle (deriving the formula)

    This is the question. The following is the solutions I found: I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how...
  11. S

    Finding the Determinant to find out if the matrix is invertible

    question: My first attempt: my second attempt: So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...
  12. K

    MHB Proving or disproving this matrix V is invertible.

    Hello everyone, I find an interesting matrix which seems to be always invertible. But I have no idea how to prove it! So I write down here for some ideas. Here is the problem: Let us take $n\in \mathbb{N}^*$ bins and $d\in \mathbb{N}^*$ balls. Denote the set $B = \{\alpha^1, \ldots...
  13. K

    MHB Proving or disproving this matrix V is invertible.

    Let us take $n\in \mathbb{N}^*$ bins and $d\in \mathbb{N}^*$ balls. Denote the set $B = \{\alpha^1, \ldots, \alpha^m\}$ to be all possible choices for putting $d$ balls into $n$ bins, such as $$\alpha^1 = (d,0,\ldots, 0), ~ \alpha^2 = (0,d,\ldots, 0), \ldots$$ Let us define the matrix $V$ as...
  14. Isaac0427

    I Matrices Commuting with Matrix Exponential

    The summary pretty much explains my question. I know that ##\left[ A, e^B \right]=0## if ##[A,B]=0## (and can prove it), but I can't figure out how to prove if it is or is not an "if and only if" statement. Thanks in advance!
  15. M

    MATLAB Matrix multiplication without a for-loop for an uneven size matrix and a vector

    Hi PF! I am trying to multiply each component of B by the matrix A and then solve A\C. See the code below. A = rand(4); B = rand(5,1); C = rand(4,1); for i = 1:5 sol(:,i) = (B(i)*A)\C end But there has to be a way to do this without a for-loop, right? I'd really appreciate any help you have!
  16. M

    MATLAB Plotting a 3D image from a matrix for a complex domain

    Hi PF! Each element of an ##n\times m## matrix is complex valued. In the following code, I call this "domain". There is also an ##n\times m## matrix that is real valued, below I call this "f". I'd like to plot a 3D image where the ##x-y## plane is the complex plain given by the coordinates...
  17. M

    MHB Relations between map and matrix

    Hey! 😊 Let $1\leq m,n\in \mathbb{N}$ and let $\mathbb{K}$ be a field. For $a\in M_m(\mathbb{K})$ we consider the map $\mu_a$ that is defined by \begin{equation*}\mu_a: \mathbb{K}^{m\times n}\rightarrow \mathbb{K}^{m\times n}, \ c\mapsto ac\end{equation*} I have show that $\mu_a$ is a linear...
  18. brotherbobby

    On the orthogonality of the rotation matrix

    Let me start with the rotated vector components : ##x'_i = R_{ij} x_j##. The length of the rotated vector squared : ##x'_i x'_i = R_{ij} x_j R_{ik} x_k##. For this (squared) length to be invariant, we must have ##R_{ij} x_j R_{ik} x_k = R_{ij} R_{ik} x_j x_k = x_l x_l##. If the rotation matrix...
  19. ChrisVer

    Question regarding the exponential of a matrix

    Hi, I think this is a nitpicking question, but oh well let me hear your inputs. Actually I tried to solve this question straightforwardly, by Taylor expanding the exponential and showing that: \textbf{A}^n = \begin{pmatrix} a^n & nba^{n-1} \\ 0 & a^n \end{pmatrix} i.e. e^{\textbf{A}t} =...
  20. Sunny Huang

    A How to calculate the interband velocity matrix in a graphene system

    In graphene system, the velocity operator sometimes is v= ∂H/ħ∂p, and its matrix element is calculated as <ψ|v|ψ>, i.e., v_x = v_F cos(θ) and v_y = v_F sin(θ) [the results are the same with Eq. 25] for intraband velocity. Recently, I see a new way to calculate the velocity matrix (Mikhailov...
  21. T

    I Invertible Matrix Proof: A-Transpose * M * A (n by m)

    Hello Suppose if have a matrix that is purely diagonal with NO zeros: M (which is n by n -square) Suppose I have another matrix the contains coordinate information, call it A. This one is NOT a square matrix, but, (n by m) (where, in general m < n) I form this: Q = A-transpose * M * A...
  22. C

    I Matrix Representations of the Poincare Group

    I'm trying to 'see' what the generators of the Poincare Group are. From what I understand, it has 10 generators. 6 are the Lorentz generators for rotations/boosts, and 4 correspond to translations in ℝ1,3 since PoincareGroup = ℝ1,3 ⋊ SO(1,3). The 6 Lorentz generators are easy enough to find in...
  23. S

    B What are matrix transformations?

    What does a transformation of a matrix means and how it could be used?
  24. B

    MHB Creating Linearly Dependent Square Matrix from Cam Mechanism Equation

    I have an equation that comes from an especific topic of cam mechanisms and it goes like this: $$ 2M[tan(B)-B] - \beta Ntan(B) - 2\pi\sqrt{1 - N^2} = 0 \ \ \ \ \ \ \ \ \ (1) $$ For this it doesn't matter what each variable means. I'm trying to create a 3x3 matrix with a determinant equal to...
  25. karush

    MHB What is the next step in Gauss-Jordan Elimination for this augmented matrix?

    complete $$\left[ \begin{array}{rrrr|r} 1& -6& 4& 0&-1\\ 0& 2& -7& 0&4\\ 0& 0& 1& 2&-3\\ 0& 0& 4& 1&2\ \end{array}\right]$$ ok assume next step is $r_2/2$ and $r_4/4$ introducing fractions
  26. F

    A Relation between Dirac's equation density matrix and current with spin

    After computind dirac 1D equation time dependant for a free particle particle I get 2 matrixs. From both,them I extract: 1) the probablity matrix P =ps1 * ps1 + psi2 *psi2 2) the current matrix J = np.conj(psi1)*psi2+np.conj(psi2)*psi1 I think that current is related to electricity, and...
  27. C

    A Off-diagonal elements in a density matrix

    Hello everyone, I have a math / physics question that has been with me for a while. I would be grateful if someone could help me. Given a density matrix, what is the minimum value a sum of some of its off-diagonal elements can assume (or the most negative value)? Remark: if one collect an...
  28. T

    A Evaluating Matrix Spin Dependent Term in Dirac Quadratic Equation

    I derive the quadratic form of Dirac equation as follows $$\lbrace[i\not \partial-e\not A]^2-m^2\rbrace\psi=\lbrace\left( i\partial-e A\right)^2 + \frac{1}{2i} \sigma^{\mu\nu}F_{\mu \nu}-m^2\rbrace\psi=0$$ And I need to find the form of the spin dependent term to get the final expression $$g...
  29. O

    I Rotate Functions with Derivatives: A Quantum Mechanics Homework

    I was solving a problem for my quantum mechanics homework, and was therefore browsing in the internet for further information. Then I stumbled upon this here: R is the rotation operator, δφ an infinitesimal angle and Ψ is the wave function. I know that it is able to rotate a curve, vector...
  30. M

    Mathematica Running a matrix of integrals in parallel

    Hi PF! I am trying to computer a matrix of integrals. Think of it something like this: Table[Integrate[x^(i*j), {x, 0, 1}], {i, 0, 5}, {j, 0, 5}] I have 16 cores, and would like to have each core handle a specified amount of integrals. Anyone know how to do this? Thanks so much!
  31. Avatrin

    A Numerically solving matrix Riccati ODE

    Hi I need to solve an equation of the form $$\dot{X}(t) = FX(t) + X(t)F^T + B$$ All of these are matrices. I have an initial condition X(0)=X_0. However, I have no idea how to proceed. How can I make any progress?
  32. H

    Matrix and cross section of the Xf -> Xf process

    The attempt at a solution follows as an attachment. If you are only interested in the solution without the derivation, look at page 1 and page 9.
  33. M

    I Weird covariance matrix for linear fit

    Hello! I really don't know much about statistics, so I am sorry if this questions is stupid or obvious. I have this data: ##x = [0,1,2,3]##, y = ##[25.885,26.139,27.404,30.230]##, ##y_{err}=[1.851,0.979,2.049,6.729]##. I need to fit to this data the following function: $$y = a (x+0.5)/4.186 +...
  34. J

    Rotation by matrix multiplication -- confirmation please

    The below matrix represents a rotation. 0 0 -1 0 1 0 1 0 0 Im trying to obtain the general point (x y z) when rotated by the above rotation matrix? So visited https://www.andre-gaschler.com/rotationconverter/ entered the above figures and not sure which entry would be x y z but assume it...
  35. dykuma

    Hermitian Matrix and Commutation relations

    I think I roughly see what's happening here. > First, I will assume that AB - BA = C, without the complex number. >Matrix AB equals the transpose of BA. (AB = (BA)t) >Because AB = (BA)t, or because of the cyclic property of matrix multiplication, the diagonals of AB equals the diagonals of...
  36. M

    MHB Exploring Properties of Matrix A: Unit Vectors and Eigenvalues

    Hey! :o We have the matrix $A=\frac{1}{3}\begin{pmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ 2 & -2 & 1\end{pmatrix}$. Show that there is an unit vector $v_1$, such that $A=I-2v_1v_1^T$. We consider an orthogonal matrix $Q=\begin{pmatrix}v_1 & v_2 & v_3\end{pmatrix}$. Show that...
  37. C

    I Correlations vs. negligence of correlations in a covariance matrix

    Suppose I have a model composed of two parameters ##(a,b)## that I want to describe a set of data points with. In CASE A, I fit the model taking into consideration the correlations between the data points (that is, in the chi square formulation I use the full covariance matrix for the data) and...
  38. Athenian

    Finding the Rotation Matrix for Vector Alignment

    Below is the attempted solution of a tutor. However, I do question his solution method. Therefore, I would sincerely appreciate it if anyone could tell me what is going on with the below solution. First off, the rotation of the matrix could be expressed as below: $$G = \begin{pmatrix} AB & -||A...
  39. E

    Eigenvalues of an orthogonal matrix

    I'm fairly stuck, I can't figure out how to start. I called the matrix ##\mathbf{A}## so then it gives us that ##\mathbf{A}\mathbf{A}^\intercal = \mathbf{I}## from the orthogonal bit. I tried 'determining' both sides... $$(\det(\mathbf{A}))^{2} = 1 \implies \det{\mathbf{A}} = \pm 1$$I don't...
  40. M

    MHB Projection Matrix Calculation for Vectors a_k: P_6P_5P_4P_3P_2P_1a_0

    Hey! :o We have the vectors $\displaystyle{a_k=\begin{pmatrix}\cos \frac{k\pi}{3} \\ \sin \frac{k\pi}{3}\end{pmatrix}, \ k=0, 1, \ldots , 6}$. Let $P_k$ be the projection matrix onto $a_k$. Calculate $P_6P_5P_4P_3P_2P_1a_0$. Are the elements of the projection matrix defined as...
  41. H

    I Building a coefficient matrix for a system of equations

    I want to solve the following system of equations ##M_{1} = f_1+f_2+m_1+m_2\ \ ;\ \ M_{7} = f_1+f_2+s_1+s_2\ \ ;\ \ M_{13} = m_1+m_2+s_1+s_2## ##M_{2} = f_1+f_3+m_1+m_3\ \ ;\ \ M_{8} = f_1+f_3+s_1+s_3\ \ ;\ \ M_{14} = m_1+m_3+s_1+s_3## ##M_{3} = f_1+f_4+m_1+m_4\ \ ;\ \ M_{9} = f_1+f_4+s_1+s_4\...
  42. G

    Find the matrix representation

    $$\hat{S_+} = \hbar \begin{bmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{bmatrix}$$ $$\hat{S_-} = \hbar \begin{bmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{bmatrix}$$ $$\hat{S_x} = \hbar/\sqrt{2} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0...
  43. T

    A Algebraic proof that Euler angles define a proper rotation matrix

    I have asked this question twice and each time, while the answers are OK, I am left dissatisfied. However, now I can state my question properly (due to the last few responses). Go to this page and scroll down to the matrix for sixth row of the proper Euler angles...
  44. evinda

    MHB Find Matrix A for System Ax=1, 3

    Hello! (Wave) The general solution of the system $Ax=\begin{bmatrix} 1\\ 3 \end{bmatrix}$ is $x=\begin{bmatrix} 1\\ 0 \end{bmatrix}+ \lambda \begin{bmatrix} 0\\ 1 \end{bmatrix}$. I want to find the matrix $A$. I have done the following so far: $$x=\begin{bmatrix} 1\\ 0 \end{bmatrix}+...
  45. BoltE

    How do you invert a 3D matrix? (Tensor inversion)

    I would like to solve a system of systems of equations Ax=b where A is an n x m x p tensor (3D) matrix, x is a vector (n x 1), and b is a matrix (n x p). I haven't been able to find a clear walk-through of inverting a tensor like how one would invert a regular matrix to solve a system of linear...
  46. SpaceMonkeyCaln

    [Linear Algebra] Matrix Transformations

    Evening, The reason for this post is because as the title suggests, I have a question concerning matrix transformation. These are essentially test prep problems and I am quite stuck to be honest. Here are the [questions](https://prnt.sc/riq7m0) and here are the...
  47. G

    Fast pentadiagonal matrix solver

    Hello, I'm currently writing a numerical simulation code for solving 2D steatdy-state heat conduction problems (diffusion equation). After reading and following these two book references (Numerical Heat Transfer and Fluid Flow from Patankar and And Introduction to Computational Fluid Dynamics...
  48. S

    Python: inverse of a block matrix

    I am using the following code. It's returning the block matrix (Z) raised to negative one (think about inputting 22/7 in a Casio fx-991ES PLUS). import sympy as sp from IPython.display import display X = sp.Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3]]) i = sp.Matrix([[1], [1], [1]]) Z =...
  49. M

    A Elementwise Derivative of a Matrix Exponential

    Hi all. A problem has arisen whereby I need to maximize a function which looks like $$ f(A) = \mathbf{w}^T \left[\int_0^t e^{\tau A} M e^{\tau A^T} d\tau \right]^{-1} \mathbf{w} $$ with respect to the nxn matrix A (here, M is a covariance matrix, so nxn symmetrix and positive-definite, w is an...
  50. M

    I Expressing the Matrix Transpose Function: Is There a Different Approach?

    One way to express a function of a matrix A is by a power series (a Taylor expansion). It is not too difficult to show that two functions f(A) and g(A) with such a power series representation must commute, i.e. f(A)g(A) = g(A)f(A). But matrices typically do not commute with their own transpose...
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