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. 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...
  2. 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...
  3. 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...
  4. 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 =...
  5. 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...
  6. 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...
  7. Jozefina Gramatikova

    Visualising a matrix in Python

    As part of a group project, we have been asked to create a code which produces the following matrix: I want to create the graphics of this in a way that looks similar to this: But of course with different number of cells, different colours, etc. The problem is that from what I have found...
  8. Javier2808

    I Commutator's Matrix representation

    Hello! I have checked commutator matrix form of $$\vec{p}=im/\hbar [H,\vec{x}]$$ but I realized i don't undertand something I have $$[H,\vec{x}]=H\vec{x}-\vec{x}H$$ and $$(H\vec{x})_{i}=H_{ij}x_j$$ & $$\ ( \vec{x}H)_{i}=x_jH_{ji}$$ but what is the second term matrix representation...
  9. Z

    MHB Maximizing Tr(A) & Unique Solution of Matrix A w/ Infinite Solutions

    Hello! I am new here, and I need (urgent) help regarding the following question: Let $\boldsymbol{A}_{(n\times n)}=[a_{ij}]$ be a square matrix such that the sum of each row is 1 and $a_{ij}\ge0$$(i=1,2,\dots,n~\text{and}~j=1,2,\dots,n)$ are unknown. Suppose that...
  10. N

    I Example of a Lie group that cannot be represented in matrix form?

    I am not sure if this is the right forum to post this question. The title says it all: are there examples of Lie groups that cannot be represented as matrix groups? Thanks in advance.
  11. cookiemnstr510510

    Understanding inner product space and matrix representations of Operat

    (scroll to bottom for problem statement) Hello, I am wondering if someone could break down the problem statement in simpler terms (not so math-y). I am struggling with understanding what is being asked. I will try to break it down to the best of my ability Problem statement:Consider the inner...
  12. M

    MHB Define matrix to get a row operation of type 1

    Hey! :o We have the matrices \begin{equation*}a=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}, \ \ E_{1,3}=\begin{pmatrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}, \ \ u_n=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & 1\end{pmatrix}\end{equation*} I have...
  13. jk22

    A Should tensor sum be used in matrix mechanics?

    Suppose the Bell operator ##B=|AB(1,2)+AB(1,3)+AB(2,3)|## With ##AB\in{1,-1}## Nonlocal realism implies ##B\in{1,3}## However using usual matrix sum one eigenvalues for the result of measurement can be smaller than 1, implying nonlocal realism cannot explain the quantum result. However if...
  14. Fochina

    Proof: Relationship between a linear map and the associated matrix

    Hi! I don't understand how to demonstrate the following exercise. Let ##F: R^{n} \rightarrow R^{n}## be a linear map which is invertible. Show that if ##A## is the matrix associated with ##F##, then ##A^{-1}## is the matrix associated with the inverse of ##F##.
  15. F

    MHB Solving a Matrix Problem on IGCSE Past Paper: Part B

    I have been teaching myself matrices for my IGCSE course and I ran into a problem in a past paper which I have no clue how to solve. The problem is part b of the attached image. Thanks for your help in advance.
  16. johnconner

    I Transformation matrix for an expanding space

    Hello. I am confused with this matter that how should we write the transformation matrix for an expanding space. consider a spacetime that is expading with a constant rate of a. now normally we scale the coordinates for expansion which makes the transformation matrix like this: \begin{pmatrix}...
  17. M

    MHB Unfamiliar hessian matrix expression

    I am familiar with the hessian matrix having the square in the numerator and a product of partial derivatives in the denominator: $Hessian = \frac{\partial^2 f}{\partial x_i \partial x_j}$ However, I have come across a different expression, source...
  18. M

    Stiffness Matrix Method: Symmetry vs Introducing a new node / joint

    Hi, In the question outlined in the images (apologies for the poor quality of the scans), the chosen solution has opted to use a symmetry argument and proceed from there. Question is from "Structures: theory and analysis" by Williams & Todd My question is: How could we approach the same...
  19. Kaguro

    I Spectrum of a function vs of a matrix

    The Fourier transform of a function is called its spectrum. The set of eigenvalues of a matrix is also called a spectrum. Why the same name? Is there some hidden connection between these two?
  20. JD_PM

    Matrix representation of a linear mapping

    I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##. This can be represented by a diagram: Thus note...
  21. A

    Matrix which reverses Gram-Schmidt - Linear Algebra

    My idea was to write out the formulas for the orthogonal q vectors in terms of the input vectors using the basics of gram-schmidt. Then, I would rewrite those equations suhc that the a vectors were written in terms of the q vectors. And then, try to find some matrix which would capture the...
  22. S

    Find the sampling matrix and sampling structure for R, G and B components

    Hello, everyone. :) All I could gather is that, if I'm correct, lattices are spans of the column vectors of the matrix within the "LAT()" notation and the X and Y occurrences are unit placeholders (such as the pixel unit (since this is in the context of image processing)). And, as an attempt...
  23. Wi_N

    Decide a matrix for a vector that goes through various morphs

    vector=(abc) 1. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) \\ 0& sin(\theta) & cos(\theta) \end{pmatrix}$$ The rotation part is correct. 2. $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0& 0 & 0 \end{pmatrix}$$ is wrong apparently how do I do the mirroring? step 3 i can do...
  24. Athenian

    I Lorentz Boosts: Finding Speed, Coordinates & Rotation w/ Matrix Multiply

    Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening: $$ \left( \begin{array}{cccc} \frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\ 0 & 1 & 0 & 0...
  25. L

    Finding Cartan Subalgebras for Matrix Algebras

    This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...
  26. Zouatine

    Stiffness Matrix For a Beam

    hello i hope everyone doing well, I have problem in Stiffness Matrix For Beam element (2 nodes ) i have a beam element i want to get a stiffness matrix: we have beam element (2 nodes) node (1) : u1 horizontal displacement, v1 vertical displacement node (2): u2 horizontal displacement , v2...
  27. H

    I Matrix Representation of the Angular Momentum Raising Operator

    In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem. This should be value...
  28. Monoxdifly

    MHB [ASK] Determinant of a Matrix with Polynomial Elements

    Help me if what I have done so far can be simplified further.
  29. N

    How to judge the singularity of a matrix in numerical method?

    Summary: different methods give different results. They are not consistent. Summary: different methods give different results. They are not consistent. I use two different methods to detect whether a matrix is singular. The result of calculating the determinant of a 9-order square matrix is...
  30. A

    Density Matrix of Unpolarized Light

    I have unfortunately no attempt for 31.i. I don't know how to start the problem. 31.ii is attached. Can someone give me a tip on how to start 31.i? I also have troubles solving 29.iii. I have attached that as well.
  31. D

    Find n and m values of a linear Transformation if given a matrix A

  32. lekh2003

    B Matrix Exponential: Researching Origins & Applications

    I am conducting research into the matrix exponential, and I would like to discuss the mathematical development of the exponential, and eventually some applications in terms of quantum physics. Currently, my problem is tracking down the original usage of this technique. I am trying to find a...
  33. Diracobama2181

    I Possible simple Density Matrix Question

    If I am given that the density matrix of an incoming beam of spin 1 particles of the form $$p_o=(1/3)[|1><1|+|0><0|+|-1><-1|]$$, aand I needed to find the fraction of particles that would be found with a spin x component of zero, how would I go about solving this problem? My hunch is that I...
  34. evinda

    MHB How to Sort Elements of a Matrix in O(n) Time Complexity?

    Hello! (Wave) Let a matrix with $n$ integers in the interval $[1,100]$. I want to write an algorithm that sorts the elements of the matrix in $O(n)$. [Hint: Count and use the number of times that each number appears] I have thought the following: We create a matrix with $100$ entries. If the...
  35. S

    I Find matrix of linear transformation and show it's diagonalizable

    The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...
  36. J

    Quaternions and Direction Cosine Matrix changing in time

    I've already posted this question on the mathematics website of stack exchange, but I've received more help here in the past so will share it here as well. I am developing a tool for missile trajectory (currently without guidance). One issue is that the aerodynamic equations on the missile are...
  37. Q

    A Solutions of Matrix Equation A X B^T = C

    I know that every ##m×n## matrix ##D## can be expressed in the form $$P_1DP_2 = LU$$ where ##P_1## and ##P_2## are permutation matrices, ##L## is a unit lower triangular matrix, and ##U## has the form $$\begin{bmatrix} U_1 & U_2 \newline 0 & 0 \newline \end{bmatrix}$$ where ##U_1## is an...
  38. christang_1023

    Decompose Involutory Matrix into Difference of Two Idempotents

    I feel confused about proving the two terms are idempotents.
  39. T

    Simplifying a matrix algebra equation (revised)

    I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting...
  40. R

    MHB Invertiable Matrix - explaining it to children - ideas how to teach

    How can I make a curriculum to student, to explain the term Invertiable Matrix? What would I use to explain? [tools, whiteboard, papers. printed website pages...] How can I order the curriculum? What will be in the start of year and what will be next? This new material that I need to teach in...
  41. maistral

    A Solve Weird Matrix Equation to Get Values for P

    So while I was trying to browse some notes with regard to Laplace transforms in solving systems of ODE, this matrix came up: I could easily just use RK4 and call it a day to nuke the systems of ODE, but this actually made me curious. Apparently one can transform the matrix DE into another form...
  42. Wrichik Basu

    Python Transpose of a non-square matrix (without using ndarray.transpose)

    While the prefix of the thread is Python, this could be easily generalised to any language. It is absolutely not the first time I am working with an array, but definitely the first time I am facing the task of defining the transpose of a non-square matrix. I have worked so much with arrays in...
  43. S

    Showing a matrix A is diagonalisable

    Show that ##\{\mathbf{v}_1\}## is linearly independent. Simple enough let's consider $$c_1\mathbf{v}_1 = \mathbf{0}.$$ Our goal is to show that ##c_1 = 0##. By the definition of eigenvalues and eigenvectors we have ##A\mathbf{v}_1= \lambda_1\mathbf{v}_1##. Let's multiply the above equation and...
  44. Jamister

    I What is really that density matrix in QM?

    Summary: What are the basic assumptions of QM about the density matrix? The subject of density matrix in quantum mechanics is very unclear to me. In the books I read (for example Sakurai),they don't tell what are the basic assumptions and how you derive from them the results of the density...
  45. Parzeevahl

    Comp Sci Multiplication of two 2x2 matrices in Fortran

    I have tried to do this using arrays and do loops: program matrixmul implicit none real A(2, 2), B (2, 2), C (2, 2) integer i, j, k write (*, *) 'Input: First matrix' do i = 1, 2 do j = 1, 2 read (*, *) A (i, j) enddo enddo write (*, *) 'Input: Second...
  46. karush

    MHB 5.3 Show that a square matrix with a zero row is not invertible.

    Show that a square matrix with a zero row is not invertible. first a matrix has to be a square to be invertable if $$\det \begin{pmatrix}1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}=1$$ then $$\begin{pmatrix} 1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}^{-1} =\begin{pmatrix}1&0&0\\ 0&1&0\\ -3&0&1...
  47. F

    Python Invert a matrix from a 4D array : equivalence or difference with indexes

    I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...
  48. L

    The Matrix and quantum mechanics

    I am fascinated again with The Matrix especially as it could pertain to the minimum instrumental version of quantum mechanics. Read this conversation (the quantum stuff added) The Matrix is also the minimum instrumental version of quantum mechanics which "explains" our daily world...
  49. A

    I Understanding Operators in Matrix Mechanics

    I'm trying to understand some notes that I have been given on Matrix Mechanics, specifically how the matrix element comes about and builds a matrix which when used applies the effect of an operator on a wavefunction. But I'm having some difficulties following what's being done in the notes with...
  50. M

    Mathematica Adding coordinates to a matrix

    Hi PF! Given a matrix and vector $$ \begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix},\\ \begin{bmatrix} 1\\ 2 \end{bmatrix} $$ how can I merge the two to have something like this $$ \begin{bmatrix} (1,a) & (1,b) & (1,c)\\ (2,d) & (2,e) & (2,f) \end{bmatrix} $$
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