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. S

    I Consistent matrix index notation when dealing with change of basis

    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...
  2. P

    A Purification of a Density Matrix

    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...
  3. James1238765

    I Evaluating the quark neutrino mixing matrix

    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} \\...
  4. Mayhem

    B Operator that returns unique number of binary matrix

    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...
  5. C

    Spectral decomposition of 4x4 matrix

    ## 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...
  6. yucheng

    I Partial trace and the reduced density matrix

    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...
  7. E

    I From Einstein Summation to Matrix Notation: Why?

    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...
  8. chwala

    Express the given matrix in the form ##L_1DU_1##

    $$\begin{bmatrix} 4& 3 \\ 6 & -2 & \\ \end{bmatrix}= \begin{bmatrix} 1& 0 \\ a& 1& \\ \end{bmatrix}⋅ \begin{bmatrix} b& c \\ 0& d& \\ \end{bmatrix}$$ ##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 & \\...
  9. T

    I Getting eigenvalues of an arbitrary matrix with programming

    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...
  10. A

    I Determining elements of Markov matrix from a known stationary vector

    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...
  11. atyy

    I Are we living in the matrix? No.

    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...
  12. J

    Transformation Matrix T in Terms of β1, β2 with Row Reduction Explained

    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.
  13. A

    Prove the identity matrix is unique

    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...
  14. dRic2

    I Transform a 2x2 matrix into an anti-symmetric matrix

    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...
  15. V

    Gaussian Elimination of Singular Matrix with partial pivoting

    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...
  16. Ashish Somwanshi

    Matrix representation in QM Assignment -- Need some help please

    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...
  17. Graham87

    Quantum Mechanics - Matrix representations

    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!
  18. shahbaznihal

    A Computing the Fisher Matrix numerically

    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...
  19. K

    A Matrix representation of a unitary operator, change of basis

    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...
  20. S

    Can every symmetric matrix be a matrix of inertia?

    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)...
  21. H

    Prove that every unitary matrix is diagonalisable by a unitary matrix

    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...
  22. G

    I Diagonal Matrix of Stress-Energy Tensor: Why?

    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...
  23. A

    I Changing diagonal elements of a matrix

    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...
  24. A

    Is this vector in the image of the matrix?

    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...
  25. B

    C# How close to Gaussian a 2D Matrix percentage is in C#

    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%:
  26. H

    Find a matrix ##C## such that ##C^{-1} A C## is a diagonal matrix

    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...
  27. C

    Transition Rate Matrix for 5 Processing Units

    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...
  28. C

    Transition matrix of a paint ball game

    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...
  29. Haorong Wu

    Solutions of first-order matrix differential equations

    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...
  30. topsquark

    MHB Matrix Methods for Difference Problems II

    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...
  31. topsquark

    MHB Matrix Methods for Difference Problems

    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...
  32. H

    Vectors as geometric objects and vectors as any mathematical objects

    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...
  33. Jamister

    I Proof about a positive definite matrix

    I need to prove the following: A symmetric real matrix ##A## with positive elements ##a_{i,j}\geq 0## can’t be definite positive matrix (i.e. with only positive eigenvalues) if the following condition holds: $$\sum_{i=1}^{N-1}a_{i,i+1}>\frac{1}{2}\sum_{i=1}^{N}a_{i,i}=\frac{1}{2}\text{Tr}(A)$$...
  34. H

    I How can I convince myself that I can find the inverse of this matrix?

    If I have a ##n\times n## matrix $$ U= \begin{bmatrix} u_{11} & u_{12} &u_{13} & \cdots u_{1n} \\ 0 & u_{22} & u_{23} & \cdots u_{2n} \\ 0&0 &u_{33} &\cdots u_{3n}\\ \vdots & \vdots &\vdots & \cdots \vdots \\ 0 & 0 & 0 &\cdots u_{nn} \end{bmatrix} $$ Now, I don't want to use the fact that it's...
  35. murshid_islam

    I Matrix representation for closed-form expression for Fibonacci numbers

    From the wikipedia page for Fibonacci numbers, I got that the matrix representation for closed-form expression for Fibonacci numbers is: \begin{pmatrix} 1 & 1 \\ 1 & 0\\ \end{pmatrix} ^ n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1}\\ \end{pmatrix} That only works...
  36. GingFritz

    How determine that a matrix is a tensor?

    I have the matrix $$ A = \left(\begin{array}{cc} y^2 & -xy\\ -xy & x^2 \end{array} \right) $$ I know that to prove that the matrix is a tensor, it transform their elements in another base. But I still without how begin this problem. Help please! Thanks.
  37. TheScienceAlliance

    MHB Matrix: True or False? AB=BA

    If A and B are matrices that AC = AC and BC=CB, where C is a matrix whose first row's entries are 0 1 and the second row's entries are -1 0, then AB=BA.
  38. Salmone

    I Matrix element in problem with hydrogen atom

    I have a problem in calculate a matrix element in a problem with hydrogen atom. I have an hydrogen atom and Hamiltonian eigenstates ##|n,l,m>## where ##n## are energy quantum numbers, ##l## are ##L^2## quantum numbers and ##m## are ##L_z## quantum numbers, I have to calculate the matrix element...
  39. M

    Is result of vector inner product retained after matrix multiplication?

    Hi, I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea. Question: Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve...
  40. Rlwe

    I Determinant of a specific, symmetric Toeplitz matrix

    Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...
  41. J

    The infinite limits of the probability transition matrix for Markov chain

    Consider a Markov chain with state space {1, 2, 3, 4} and transition matrix P given below: Now, I have already figured out the solutions for parts a,b and c. However, I don't know how to go about solving part d? I mean the question says we can't use higher powers of matrices to justify our...
  42. P

    Question about hollow matrix and diagonalization

    A quick and simple question. I just realized that this has been posted in the wrong section, but ill give it a try anyway. Does anyone know if it's possible to diagonalize a hollow matrix? What i mean by a hollow matrix is a matrix with only zero entries along the diagonal.
  43. Arquimedes

    A Schmidt decomposition - How do I find the matrix related to the state?

    Hello, I am currently studying the Schmidt decomposition and how to use it to determine if a state is entangled or not and I can't understand how to write the state as a matrix so I can apply the Singular Value Decomposition and find the Schmidt coefficients. The exercise I am trying to complete...
  44. P

    Spin matrix representation in any arbitrary direction

    I've tried to use the 1st equation as a matrix to determine, but it clearly isn't a diagonal matrix. My guess is that I need to find the spin matrix along the direction ##\hat{n}##, but do I need to find the eigenstates of ##\sigma \cdot \hat{n}## first and check if they form a diagonal matrix...
  45. T

    B Does this kind of square matrix exist?

    I had a homework question that gives A as an arbitrary matrix. Then the question states that A^2=A Now I manipulate the equation to give this A^2-A=0. -->A(A-I)= 0 So A can be I or 0 Are there any other values A can take?
  46. H

    Find the basis so that the matrix will be diagonal

    First of all, it is clear that we can find such a bases (the theorem is given in almost all of the books, but if you want to share some insight I shall be highly grateful.) We can show that ##W## will be the set of all real polynomials with degree ##\leq 2##. So, let's have ##\{1,x,x^2\}## as...
  47. S

    Comp Sci I can't get this task right regarding a simple ' 2D game' as a matrix

    I have also put some notes on what is to be done in the problematic function Note also that this is not homework, but am just preparing for an exam. Thanks in advance! forgot to provide the code here, so here it is: # RPG subsystem: check whether the next player move on a 5x5 tileset is...
  48. Eclair_de_XII

    B Determinant of a transposed matrix

    By definition, ##\det A=\sum_{p_j\in P}\textrm{sgn}(p_j)\cdot a_{1j_1}\cdot\ldots\cdot a_{nj_n}##, where ##P## denotes the set of all permutations of the ordered sequence ##(1,\ldots,n)##. Denote the number of permutations needed to map the natural ordering to ##p_j## as ##N_j##. Now consider...
  49. J

    Very High Standard Deviation in Excitation Emission Matrix Measurement

    Hi, I obtain really high standard deviations in Excitation-Emission Spectra mainly for the phenolic compounds in olive oil (Em: 290-350nm). Method: I weigh 0.05g of olive oil and dilute it up to 25ml with cyclohexane to remain in the range of linearity for absorbance measurements to correct...
  50. Eclair_de_XII

    I'm not able to write a script that reduces a matrix to RREF in Python

    Summary:: I'm not asking for help, but I'm asking for an opinion. Is this a sign that I probably should not be pursuing a career in software development or computer science? I basically feel like this in general wrt any subject I am studying, really, whenever I feel stumped on a given problem...
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