Matrix Definition and 1000 Threads

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. L

    I 1x1 Matrix is a scalar?

    I've seen dot product being represented as a (nx1 vector times a (mx1)^T vector. This gives a 1x1 matrix, whereas the dot product should give a scalar. I have found some threads online saying that a 1x1 matrix IS a scalar. But none of them seem to answer this question: you can multiply a 2x2...
  2. N

    I In 4x4 matrix when does row swapping not affect eigenvaluess?

    The title pretty much says it... I know that in general eigenvalues are not necessarily preserved when matrix rows or columns are swapped. But in many cases it seems they are, at least with 4x4 matrices. So is there some specific rule that says when eigenvalues are preserved if I swap two rows...
  3. Kostik

    A Making sense of Dirac's rotation operator in "General Theory of Relativity"

    In Dirac's "General Theory of Relativity", Chap. 34 on the polarization of gravitational waves, he introduces a rotation operator ##R##, which appears to be a simple ##\pi/2## rotation, since $$R \begin{pmatrix} A_0 \\ A_1 \\ A_2 \\ A_3 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 &...
  4. S

    I Matrix diagonalization algorithm

    I need to compute the vibrational frequencies of a molecule when the matrix of force constants (second derivative of the energy by the Cartesian coordinates) is provided. For such computation, this matrix must be diagonalized. Here is an example of a matrix which must be diagonalized: Here...
  5. adf89812

    I How to do Magnus expansion for time-dependent companion matrix?

    For example, consider the following system of 2 first order ODEs: $$ \left\{\begin{array}{l} x_1^{\prime}=2 t x_1+t^2 x_2 \\ x_2^{\prime}=t^3 x_1+4 t x_2 \end{array}\right. $$ This is a linear homogeneous system of 2 first order ODEs with $$A(t)=\left[\begin{array}{ll}2 t & t^2 \\ t^3 & 4...
  6. A

    Finding Non-Trivial Solution(s) For 3x3 Matrix

    I didn't have any good way to put this in the homework statement, but this is what the question is asking: For what c-value(s) will there be a non-trivial solution: ## x_1 - x_2 + x_3 = 0 ## ## 2x_1 + x_2 + x_3 = 0 ## ##-x_1 + (c)x_2 + 2x_3 = 0 ## I have spent a good couple hours looking at...
  7. G

    I How does one find the dual of a matrix?

    How does one find the dual of a matrix? Thanks.
  8. M

    I Matrix elements of Hamiltonian include Dirac delta term

    Currently reading a textbook on non-equilibrium green's functions and I'm stuck in chapter 1 where it recaps just general quantum mechanics because of dirac deltas included in the matrix elements of a generalized hamiltonian. The textbook gives this: I just don't understand how to think about...
  9. M

    Fundamental matrix linear system equivalent to linear matrix system

    My working is , Consider case where the there are two linearly independent solutions ##x'(t) = c_1x' + c_2y' = A(c_1x + c_2y)## ##(x'~y')(c_1~c_2)^T = A(x~y)(c_1~c_2)^T## Then cancelling coefficient matrix I get, ##(x'~y')= A(x~y)## ##Φ'(t) = AΦ(t) ## by definition of 2 x 2 fundamental matrix...
  10. Hennessy

    I Qubits state calculation

    Hello All, 2nd year undergrad taking my first course in modern physics. We have been given this question in a mock exam and at the bottom is the solution. When looking at a general cubit it seems the argument of both sin and cos functions should be (pi/2) not (pi/3). I have figured out a...
  11. M

    Solving system of differential equations using matrix exponential

    For this problem, The solution is, However, can someone please explain to me where they got the orange coefficient matrix from?It seems different to the original system of the form ##\vec x' = A\vec x## which is confusing me. Thanks!
  12. T

    Critical points of matrix

    My attempt is: Condition for critical point is ##x' = y' = 0##, ##0 = x - 2y \implies 2y = x## ##-2x + dy = 0## Then ##-4y + 4y = 0## However, this means that critical points are ##(2y, y)## as system is linearly dependent (both equations are the same) where ##y \in \mathbb{R}##. However, that...
  13. T

    Eigenvalue of matrix proof by induction

    We consider base case (##n = 1##), ##B\vec x = \alpha \vec x##, this is true, so base case holds. Now consider case ##n = 2##, then ##B^2\vec x = B(B\vec x) = B(\alpha \vec x) = \alpha(B\vec x) = \alpha(\alpha \vec x) = \alpha^2 \vec x## Now consider ##n = m## case, ##B^m\vec x = B(B^{m - 1}...
  14. T

    Linear Algebra Determinant proof

    I have a doubt about this problem. (a) Show that a matrix ##\left(\begin{array}{ll}e & g \\ 0 & f\end{array}\right)## has determinant equal to the product of the elements on the leading diagonal. Can you generalize this idea to any ##n \times n## matrix? The first part is simple, it is just ef...
  15. BerriesAndCream

    I Can I visualize O(3) \ SO(3) in some way?

    Hello. I know that a 3×3 orthogonal matrix with determinant = 1 (so a 3×3 special orthogonal matrix) is a rotation in 3D. I was wondering if there is a 3×3 orthogonal matrix with determinant = –1 could be visualised in some way. Thank you!
  16. R

    I Microring resonator matrix

    Hello everyone, A simple ring resonator with a bus waveguide is described by: $$ \begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ -k^* & t^* \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix} $$ I do not understand though why we have -k* and t*? Shouldn't...
  17. chwala

    Find Matrix P and the Diagonal Matrix D

    For part (b) i was able to use equations to determine the eigenvectors; For example for ##λ =6## ##12x +5y -11z=0## ##8x-4z=0## ##32x+10y-26z=0## to give me the eigen vector, ##\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}## and so on. My question is to get matrix P does the arrangement of...
  18. M

    Classification of Equlibrium Points

    I hope this is more properly laid out? We previously established that the stationery points were (1,1) and (-1,1) For this first stage I now need to create the elements of a Jacobian maitrix using partial differentation. I am confused by reference to the chain rule. Am I correct that for dx/dt...
  19. docnet

    Help understanding the definition of positive semidefinite matrix

    Please confirm or deny the correctness of my understanding about this definition. For a given set of ##t_i##s, the matrix ##(B(t_i,t_j))^k_{i,j=1}## is a constant ##k\times k## matrix, whose entries are given by ##B(t_i,t_j)## for each ##i## and ##j##. The the 'finite' in the last line of the...
  20. docnet

    Solving a separable matrix ODE.

    I have never solved a matrix ODE before, and am wondering if solving it is similar to solving ##y'=ay## where ##a## is a constant and ##y:\mathbb{R} \longrightarrow \mathbb{R}## is a function. The solution is right according to wikipedia, and I am just looking for your inputs. Thanks...
  21. docnet

    Solving a first order matrix differential equation

    Let X be a continuous-time Markov chain that hops between two states ##\{1, 2\}## with rates ##\lambda, \mu>0##, so its generator is $$Q = \begin{pmatrix} -\mu & \mu\\ \lambda & -\lambda \end{pmatrix}.$$ Solve ##\pi Q = 0## for the stationary distribution, and verify that...
  22. Filip Larsen

    I Solving matrix commutator equations?

    I have the matrix relationship $$C = A^{-1} B^{-1} A B$$ I want to solve for ##A##, where ##A, B, C## are 4x4 homogeneous matrices, e.g. for ##A## the structure is $$A = \begin{pmatrix} R_A & \delta_A \\ 0 & 1 \end{pmatrix}, A^{-1} =\begin{pmatrix} R_A^\intercal & -R_A^\intercal\delta_A \\ 0 & 1...
  23. N

    I Is there always a matrix corresponding to eigenvectors?

    I tried to find the answer to this but so far no luck. I have been thinking of the following: I generate two random vectors of the same length and assign one of them as the right eigenvector and the other as the left eigenvector. Can I be sure a matrix exists that has those eigenvectors?
  24. T

    I Correlation Matrix of Quadratic Hamiltonian

    I am struggling to rederive equations (61) and (62) from the following paper, namely I just want to understand how they evaluated terms like ##\alpha\epsilon\alpha^{T}## using (58). It seems like they don't explicitly solve for ##\alpha## right?
  25. Z

    A How to find branching ratio after modelling Lagrangian?

    hi, we have learned that after modelling Lagrangian and extracting Feynman rules from it - we can find matrix element - from which decay width can be calculated - and than Branching ratio - my question is can we use some other way of calculatiing BR , or can we use our Lagrangain in our Euler...
  26. Euge

    POTW Inequality of Determinants

    Let ##M## be a real ##n \times n## matrix. If ##M + M^T## is positive definite, show that $$\det\left(\frac{M + M^T}{2}\right) \le \det M$$
  27. F

    I Is there a better way to calculate time-shifted correlation matrices?

    Hello everyone. I have four thermometers which measure the temperature in four different positions. The data is distributed as a matrix, where each column is a sensor, and each row is a measurement. All measurements are made at exactly the same times, one measurement each hour. I have...
  28. G

    I Transfer rank2 tensor to a new basis

    The trace of the sigma should be the same in both new and old basis. But I get a different one. Really appreciate for the help. I’ll put the screen shot in the comment part
  29. G

    A Lagrangian: kinetic matrix Z_ij and mass matrix k_ij

    Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij? Quite confused about understanding this paragraph, can anybody explain it more easily?
  30. Euge

    POTW Positive Definite Block Matrices

    Suppose ##A## and ##B## are positive definite complex ##n \times n## matrices. Let ##M## be an arbitrary complex ##n \times n## matrix. Show that the block matrix ##\begin{pmatrix} A & M\\ M^* & B\end{pmatrix}## is positive definite if and only if ##M = A^{1/2}CB^{1/2}## for some matrix ##C## of...
  31. T

    B Question on basic linear algebra (new to the subject)

    It would be nice if someone could find the history of why we use the letters i and j or m and n for the basics when working with Matrices ( A = [aij]mxn ). I tried looking up the information and I was not successful. I understand what they represent in the context of the matter, but not why they...
  32. S

    B Elementary column operations and change-of-basis

    A blog post by Evan Chen https://blog.evanchen.cc/2016/07/12/the-structure-theorem-over-pids/ says that elementary row and column operations on a matrix can be interpreted as a change-of-basis. I assume this use of the phrase "change of-basis" refers to creating a matrix that uses a different...
  33. Euge

    POTW Solution to a Matrix Quadratic Equation

    Let ##A## be a complex nilpotent ##n\times n##-matrix. Show that there is a unique nilpotent solution to the quadratic equation ##X^2 + 2X = A## in ##M_n(\mathbb{C})##, and write the solution explicitly (that is, in terms of ##A##).
  34. L

    A Can a Non-Diagonal Hermitian Matrix be Diagonalized Using Unitary Matrix?

    Every hermitian matrix is unitary diagonalizable. My question is it possible in some particular case to take hermitian matrix ##A## that is not diagonal and diagonalize it UAU=D but if ##U## is not matrix that consists of eigenvectors of matrix ##A##. ##D## is diagonal matrix.
  35. DuckAmuck

    A Anti-symmetric tensor question

    The sigma tensor composed of the commutator of gamma matrices is said to be able to represent any anti-symmetric tensor. \sigma_{\mu\nu} = i/2 [\gamma_\mu,\gamma_\nu] However, it is not clear how one can arrive at something like the electromagnetic tensor. F_{\mu\nu} = a \bar{\psi}...
  36. M

    Notation for changing rows in a matrix

    For this, What was wrong with the notation I used for showing that I has swapped the rows? The marker put a purple ? Any help greatly appreciated! Many thanks!
  37. S

    I A wonderful flow chart for taxonomy of matrices

    https://upload.wikimedia.org/wikipedia/commons/d/d1/Taxonomy_of_Complex_Matrices.svg
  38. M

    Understanding Eigenvalues of a Matrix

    For this, I am confused by the second line. Does someone please know how it can it be true since the matrix dose not have an inverse. Many thanks!
  39. PhysicsRock

    Square of orthogonal matrix vanishes

    I found a the answer in a script from a couple years ago. It says the kinetic energy is $$ T = \frac{1}{2} m (\dot{\vec{x}}^\prime)^2 = \frac{1}{2} m \left[ \dot{\vec{x}} + \vec{\omega} \times (\vec{a} + \vec{x}) \right]^2 $$ However, it doesn't show the rotation matrix ##R##. This would imply...
  40. M

    Finding ##A^{-1}## of a matrix given three submatrices

    For this problem, Find ##A^{-1}## given, The solution is, However, in the first image, why are we allowed to put together the submatrices in random order? In general does someone please know why we are allowed to decompose matrices like this? Many thanks!
  41. M

    Proof of ##M^n## (matrix multiplication problem)

    For, Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n## Many thanks!
  42. M

    Diagonalizing a Matrix: Understanding the Process and Power of Matrices

    For this, Dose someone please know where they get P and D from? Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k? Many thanks!
  43. F

    Intro to Linear Algebra - Nullspace of Rank 1 Matrix

    The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
  44. nomadreid

    I Want to understand how to express the derivative as a matrix

    In https://www.math.drexel.edu/~tolya/derivative, the author selects a domain P_2 = the set of all coefficients (a,b,c) (I'm writing horizontally instead off vertically) of second degree polynomials ax^2+bx+c, then defines the operator as matrix to correspond to the d/dx linear transformation...
  45. N

    Engineering Minimum Hamming Distance for Parity Check Matrix

    My answer: Then, if I am not mistaken, the solution made in that video is mostly guessing about which columns combination can be equals to zero and I found 1st, 2nd, and 3rd rows as well as 2nd, 3rd, 4th rows are equals to zero so the minimum hamming distance is 3 since my answer is mostly...
  46. M

    Why can't we define an eigenvalue of a matrix as any scalar value?

    For this, Dose anybody please know why we cannot say ##\lambda = 1## and then ##1## would be the eigenvalue of the matrix? Many thanks!
  47. V

    Is it ok to assume matrices A and B as identity matrix?

    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 =...
  48. M

    Proving inverse of a 2 x 2 matrix is really an inverse

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

    Transformations to both sides of a matrix equation

    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...
  50. Euge

    POTW Comparing Rank and Trace of a Matrix

    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?
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