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. PainterGuy

    Left and right inverses of a non-square matrix

    Hi, It's actually not a homework problem but I still decided to post it here. Problem: Consider Ax=y, where A is mxn and has rank m. Is (A′A)⁻¹A′y a solution? If not, under what condition will it be a solution? Is A′(AA′)⁻¹y a solution? The given solution is: Consider Ax=y with A mxn and...
  2. PainterGuy

    Calculating the nth power of a matrix

    Mentor note: Since the technique used here involves differentiation, I moved this to the Calculus section. Hi, I was trying to do the problem below. I was following the approach presented in this answer. I assume the approach is correct. The answer I ended up with is clearly wrong. Where am I...
  3. LCSphysicist

    Check the spectral theorem for this matrix

    I found three projection operators $$P_{1}= \begin{pmatrix} 1/2 & & \\ & -\sqrt{2}/2 & \\ & & 1/2 \end{pmatrix}$$ $$P_{2}= \begin{pmatrix} 1/2 & & \\ & \sqrt{2}/2 & \\ & & 1/2 \end{pmatrix}$$ $$P_{3}= \begin{pmatrix} -1/\sqrt{2} & & \\ & & \\ & & 1/\sqrt{2} \end{pmatrix}$$ From this five...
  4. K

    ABCD matrix formalism for concave mirror

    Hello! I need to calculate the ABCD matrix for a thick concave mirror, in the situation in which the light comes from the plane side of the mirror, and it is the concave part that is coated (for reference, I have a Fabry Perot cavity with 2 concave mirrors, and I want to mode match the laser...
  5. PainterGuy

    Find the rank of this 3x3 matrix

    Hi, I was trying to find the rank of following matrix. I formed the following system and it seems like all three columns are linearly independent and hence the rank is 3. But the answer says the rank is '2'. Where am I going wrong? Thanks, in advance!
  6. M

    A Solve B^3 = A^2 Matrix 2x2 on C

    i know that there is the Cayley -Hamilton theorem but i don't know if i can use it and how.Do you have any ideas about it?Please give me any help.
  7. karush

    MHB T20 Suppose that A is a square matrix of size n and ......

    https://drive.google.com/file/d/1g7fjWAUEpOo2NukqFqZI4Wrujud6sjbn/view?usp=sharing $\tiny{4.288.T20}$ Suppose that A is a square matrix of size n and $\alpha \in \CC$ is $\alpha$ scalar. Prove that $\det{\alpha A} = \alpha^n\det{A}$. Using $\alpha=5$ $\det{5A}=\det\left(5\left[...
  8. H

    Find the eigenvalues of a 3x3 matrix

    Hi, I have a 3 mass system. ##M \neq m## I found the forces and I get the following matrix. I have to find ##\omega_1 , \omega_2, \omega_3## I know I have to find the values of ##\omega## where det(A) = 0, but with a 3x3 matrix it is a nightmare. I can't find the values. I'm wondering if...
  9. A

    I General worked out solution for diagonalizing a 4x4 Hermitian matrix

    Hello, I am looking for a worked out solution to diagonalize a general 4x4 Hermitian matrix. Is there any book or course where the calculation is performed? If not, does this exist for the particular case of a traceless matrix? Thank you!
  10. tanaygupta2000

    Matrix formulation of an operator

    I have successfully found the N by N matrix corresponding to the operator R. But the problem is, whenever I try to operate R on |bj> basis vectors, I am not getting |b(j+1)> as it should be. Instead, I am getting result as given in the question only by <bj|R = <b(j+1)| Matrix is not working...
  11. Tinkermen

    Mathematically determine a Lotto Draw Pattern Matrix

    Hi everyone I am new to this or any online math board community, I’m looking for assistance in determining and calculating the size of a lottery draw pattern matrix, using simple mathematics formulas. That I myself can learn to use to include math formulas on the information pages of my new...
  12. M

    MHB M has the unit matrix at the upper left side and zero everywhere else

    Hey! :giggle: Let $\lambda\in \mathbb{R}$ and \begin{equation*}a=\begin{pmatrix}1 & 2 &-1& \lambda & -\lambda \\ 0 & 1 & -1& \lambda & 2\\ 2 & 2 & 1 & 1 & 3\lambda-1 \\ 1 & 1 & 1 & \lambda & 5\end{pmatrix}\in \mathbb{R}^{4\times 5}\end{equation*} (a) Let $\lambda=1$. Determine a Basis...
  13. Y

    Help with the matrix representation of <-|+|->. Does "+"=|+>?

    Trying to use <+|+>=1=<-|-> and <-|+>=0 to prove each iteration of the equation, so I have 6 different versions to prove. But the part I'm currently stuck on is understanding how to simplify any given version. I've written out [S_x,S_y]=S_xS_y\psi-S_yS_x\psi and expanded it in terms of the...
  14. M

    MHB Give a basis to get the specific matrix M

    Hey! :giggle: We have the following linear maps \begin{align*}\phi_1:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto \begin{pmatrix}x+y\\ x-y\end{pmatrix} \\ \phi_2:\mathbb{R}^2\rightarrow \mathbb{R}, \ \begin{pmatrix}x\\ y\end{pmatrix} \mapsto...
  15. D

    I Decompose SL(2C) Matrix: Real Parameters from Complex

    Hi, suppose I am given an SL(2C) matrix of the form ##\exp(i\alpha/2 \vec{t}\cdot\vec{\sigma})## where ##\alpha## is the complex rotation angle, ##\vec{t}## the complex rotation axis and ##\vec{\sigma}## the vector of the three Pauli matrices. I would like to decompose this vector into...
  16. T

    I Meaning of Third Eigenvalue in a Tilted Ellipse in a 3x3 Matrix

    While reading the Strang textbook on tilted ellipses in the form of ax^2+2bxy +cy^2=1, I got to thinking about ellipses of the form ax^2 + 2bx + 2cxy + 2dy + ey^2=1 and wondered if I could model them through 3x3 symmetric matrices. I think I figured out something that worked for xT A x, where x...
  17. M

    Check/verify my work and answer? Pseudoinverse of matrix

    Also, if it's possible, I would really like to know the command for inputting this kind/type of problem on Ti-89 in order to check correct answers for linear algebra problems like this one.
  18. C

    Det of Triangular-like Matrix & getting stuck in Algorithmic Proof

    Find determinant of following matrix: ## A = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n-1} & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n-1} & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{n,1} & 0 & \cdots & 0 & 0 \end{pmatrix} ## Note: I tried to solve this question...
  19. S

    I Matrix Notation for potential in Schrodinger Equation

    I'm working on the time-dependent Schrodinger equation, and come across something I don't understand regarding notation, which is not specific to TDSE but the Schrodinger formalism in general. Let's say we have a non-trivial potential. There is a stage in the development of the TDSE where we...
  20. L

    A Matrix multiplication, Orthogonal matrix, Independent parameters

    Matrix multiplication is defined by \sum_{k}a_{ik}b_{kj} where ##a_{ik}## and ##b_{kj}## are entries of the matrices ##A## and ##B##. In definition of orthogonal matrix I saw \sum_{k=1}^n a_{ki}a_{kj}=\delta_{ij} This is because ##A^TA=I##. How to know how many independent parameters we have in...
  21. LCSphysicist

    How to find the determinant of this matrix?

    I think you all can see that ##a_{(i+1,j+1)} = a_{i,j} + a_{i+1,j} + a_{i,j+1}## Now the determinant always give me problem. I have and idea to reduce this matrix by Chio to a 2x2 matrix and find the determinant of this 2x2. Put i was not able to see any pattern to find what how the 2x2 matrix...
  22. james228

    A Question about Color Singlet State invariance under Unitary Matrix

  23. F

    Change of basis to express a matrix relative to a set of basis matrices

    Hello, I am studying change of basis in linear algebra and I have trouble figuring what my result should look like. From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##...
  24. S

    I Matrix construction for spinors

    I'm reading the book QFT by Ryder, in the section where ##\rm{SU(2)}## is discussed. First, he considered the group of ##2 \times 2## unitary matrices ##U## with unit determinant such that it has the form, $$U =\begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix}, \qquad \xi = \begin{bmatrix}...
  25. LCSphysicist

    Show that this column matrix is not a vector

    Summary:: I am suppose to show that this columns matrix does not transform as a vector. In another words, it is not in fact a vector. I think this become trivial if we get the rotation matrix composed of Euler angles. But, i think that it is not the best way to solve this problem, and i...
  26. karush

    MHB 311.2.2.6 use inverse matrix to solve system of equations

    $\tiny{311.2.2.6}$ Use the inverse to solve the system $\begin{array}{rrrrr} 7x_1&+3x_2&=-9\\ -2x_1&+x_2&=10 \end{array}$ the thing I could not get here without a calculator is $A^{-1}$
  27. M

    LU Factorization on Ti-89: 3x3 & 4x4 Matrix Solutions

    What's the correct command for finding an LU factorization of a 3x3 and 4x4 matrix on Ti-89 graphing calculator? I'm trying to find the correct answers and verify/check my answers for Linear Algebra problems.
  28. jk22

    I Question about density matrix

    With ##\rho=\sum_i p_i|\Psi_i\rangle\langle\Psi_i|##If the ##p_i=|\langle\Psi|\lambda_i\rangle|^2## are taken as joint probabilities given by quantum mechanics for the singlet state in EPRB then this cannot represent a statistical mix (classical) of those states because of Bell's theorem ?
  29. TheMercury79

    Is the Cube of matrix associative?

    But I actually don't get the same matrix. What I get is the transpose of the other when I change the order i.e when I do [A]^2[A] I get the transpose of [A][A]^2 and vice versa What I'm trying to do is find the cube of the expectation value of x in the harmonic oscillator in matrix form. We're...
  30. M

    MHB What is the form of that matrix?

    Hey! 😊 Calculate the Cholesky decomposition of the matrix, the only non-vanishing elements are the diagonals $1,2,3, \lambda$ and all under and upper secondary diagonal elements are $1$. For which $\lambda$ is the matrix singular? Could you please explain the form of the Matrix? Does the...
  31. archaic

    Finding a matrix from a given null space

    I have solved the exercise, so I'm not giving the vectors explicitly. I just want to know if there is a quicker way than mine. We know that ##A## must have ##4## columns and ##4## lines, and we also know that its nullity is ##2##, thus its rank is ##2##. I took the simplest matrix that can have...
  32. LCSphysicist

    Not sure about this statement in vector space and matrix

    Be ##T_{1}, T_{2}## upper and lower matrix, respectivelly. Show that we haven't matrix ##M(NxN)## such that ##M(NxN) = T_{1}\bigoplus T_{2}## I am not sure if i get what the statement is talking about, can't we call ##T_{1},T_{2} = 0##? Where 0 is the matrix (NxN) with zeros on all its entries...
  33. M

    I Definiteness of a nonsymmetric matrix

    There is no specific example but my attempt at one would be to make the non-symmetric matrix symmetric. Then we would be able the usual formulas as designed for symmetric matrices. Is this how it works? Alternatively, do I just calculate the Eigen values without making it symmetric? I don't...
  34. K

    MHB Matrix Transformation - mappings of functions

    I need to find the matrix transformation of y = \frac{1}{x} onto y = \frac{-1}{3x-1}-2 I think its \begin{bmatrix} x'\\ y' \end{bmatrix} =\begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} + \begin{bmatrix} -1\\ -2 \end{bmatrix}
  35. karush

    MHB Calculating the Inverse Matrix for a 3x3 Matrix

    $\tiny{311.2.2.31}$ $A=\left[\begin{array}{rrrrr} 1&0&-2\\-3&1&4\\2&-3&4 \end{array}\right]$ RREF with augmented matrix $\left[ \begin{array}{ccc|ccc} 1&0&-2&1&0&0 \\&&&\\-3&1&4&0&1&0 \\&&&\\ 2&-3&4&0&0&1\end{array}\right] \sim \left[ \begin{array}{ccc|ccc}1&0&0&8&3&1 \\&&&\\0&1&0&10&4&1 \\&&&\\...
  36. C

    Is is possible to multiply the matrix M with either A or c?

    is is possible to multiply the matrix M with either A or c->? And if i have to write the matrices in this form: , do i divide c-> by A or do i follow som other formula?
  37. LCSphysicist

    I Hermitian Operators and Non-Orthogonal Bases: Exploring Infinite Spaces

    The basis he is talking about: {1,x,x²,x³,...} I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the...
  38. Filip Larsen

    I Rotational invariance of cross product matrix operator

    Given that the normal vector cross product is rotational invariant, that is $$\mathbf R(a\times b) = (\mathbf R a)\times(\mathbf R b),$$ where ##a, b \in \mathbb{R}^3## are two arbitrary (column) vectors and ##\mathbf R## is a 3x3 rotation matrix, and given the cross product matrix operator...
  39. Jamister

    I Understanding Matrix Mechanics in Quantum Mechanics

    In a course of QM they mention Matrix mechanics. But what is it exactly? Is it just Heisenberg picture?
  40. B

    A Geometry of matrix Dirac algebra

    Indeed, if we take a vector field which dual to the covector field formed by the gradient from a quadratic interval of an 8-dimensional space with a Euclidean metric, then the Lie algebra of linear vector fields orthogonal (in neutral metric) to this vector field is isomorphic to the...
  41. B

    Finding the determinant of a matrix using determinant properties

    Hi, I have been having some trouble in finding the determinant of matrix A in this Q Which relevant determinant property should I make use of to help me find the determinant of matrix A and maybe matrix B also This is what I have tried for matrix A so far but it's not much help really Any...
  42. dRic2

    I Minimize grand potential functional for density matrix

    I'd like to show that, by minimizing this functional $$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$ I get the well known expression $$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$ I'm...
  43. V

    I Convert 2x2 Matrix to 1x1 Tensor

    If I have a matrix representing a 2nd order tensor (2 2) and I want to convert this matrix from M$$\textsuperscript{ab}$$ to $$M\textsubscript{b}\textsuperscript{a}$$ what do I do? I'm given the matrix elements for the 2x2 tensor. When applying the metric tensor to this matrix I understand...
  44. F

    I How to Get Final Fisher Matrix from 2 Matrices

    I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows are the same in the 2 matrixes). Now I would like to make the cross synthesis of these 2 matrixes by applying for each parameter the well known formula (coming from Maximum Likelihood Estimator...
  45. J

    I Properties of a unitary matrix

    So let's say that we have som unitary matrix, ##S##. Let that unitary matrix be the scattering matrix in quantum mechanics or the "S-matrix". Now we all know that it can be defined in the following way: $$\psi(x) = Ae^{ipx} + Be^{-ipx}, x<<0$$ and $$ \psi(x) = Ce^{ipx} + De^{-ipx}$$. Now, A and...
  46. K

    Calculating Angle Between E-Field and Current Vectors in Anisotropic Mat.

    In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector. i) Calculate the angle between the...
  47. S

    B Question about transition matrix of Markov chain

    The note I get from the teacher states that for transition matrix, the column part will be current state and the row part will be future state (let this be matrix A) so the sum of each column must be equal to 1. But I read from another source, the row part is the current state and the column...
  48. Hiero

    I Scalar powers of a matrix exponential

    Starting from the definition of a matrix exponential as a power series, how would we show that ##(e^A)^n=e^{nA}##? I know how to show that if A and B commute then ##e^Ae^B = e^{A+B}## and from this we can show that the first identity is true for integer values of n, but how can we show it’s...
  49. dontknow

    Constraints in Rotation Matrix

    In Rigid body rotation, we need only 3 parameters to make a body rotate in any orientation. So to define a rotation matrix in 3d space we only need 3 parameters and we must have 6 constraint equation (6+3=9 no of elements in rotation matrix) My doubt is if orthogonality conditions...
  50. 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...
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