# Matrix Definition and 121 Discussions

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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. ### 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...
2. ### I 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...
3. ### I Usage of First Order Elastic Constants in Soft Body Equations

Hi, I have some soft body equations that require first order elasticity constants. Just trying to figure out the proper indexing. From Finite Elements of Nonlinear Continua by J.T. Oden, the elastic constants I am trying to obtain are the first order, circled below: My particular constitutive...
4. ### I Resolve moment of inertia at an angle

Initially, I calculate the moment of inertia of of a square lamina (x-z plane). Thr this square is rotated an angle $\theta$ about a vertex and I need to calculate the new moment of inertia about that vertex. Can I split the rotated square to two squares in the x-z plane and y-z plane to find...
5. ### Finding roots and complex roots of a determinant

I need to find the values of ##\Omega## where ##(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m})(-\Omega^2 + i\gamma\Omega + \frac{2k}{3m}) - (-i\gamma\Omega)(-i\gamma\Omega) = 0## I get ##\Omega^4 -2i\gamma \Omega^3 - \frac{4k}{3m}\Omega^2 + i\frac{4k}{3m}\gamma\Omega + \frac{4k^2}{9m^2} = 0## I...
6. ### Determining value of r that makes the matrix linearly dependent

for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection. As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon...
7. ### Linear Algebra uniqueness of solution

My guess is that since there are no rows in a form of [0000b], the system is consistent (the system has a solution). As the first column is all 0s, x1 would be a free variable. Because the system with free variable have infinite solution, the solution is not unique. In this way, the matrix is...
8. ### Calculate the angle between the E-field and Current vectors in an anisotropic conductive material

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...
9. ### 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...
10. ### 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...
11. ### 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...
12. ### 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...
13. ### I Rotation of functions

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...

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### I Consequences on a system of ODEs after performing operations

Hi, I have derived a matrix from a system of ODE, and the matrix looked pretty bad at first. Then recently, I tried the Gauss elimination, followed by the exponential application on the matrix (e^[A]) and after another Gauss elimination, it turned "down" to the Identity matrix. This is awfully...
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### I Types of complex matrices, why only 3?

Hi, the three main types of complex matrices are: 1. Hermitian, with only real eigenvalues 2. Skew-Hermitian , with only imaginary eigenvalues 3. Unitary, with only complex conjugates. Shouldn't there be a fourth type: 4. Non-unitary-non-hermitian, with one imaginary value (i.e. 3i) and a...
36. S

### I Frobenius Norm of a matrix

I have calculated that a matrix has a Frobenius norm of 1.45, however I cannot find any text on the web that states whether this is an ill-posed or well-posed indication. Is there a rule for Frobenius norms that directly relates to well- and ill-posed matrices? Thanks
37. S

### I Convert complex ODE to matrix form

Hi, I have the following complex ODE: aY'' + ibY' = 0 and thought that it could be written as: [a, ib; -1, 1] Then the determinant of this matrix would give the form a + ib = 0 Is this correct and logically sound? Thanks!
38. S

### I How to check if a matrix is Hilbert space and unitary?

I have a matrix, [ a, ib; -1 1] where a and b are constants. I have to represent and analyse this matrix in a Hilbert space: I take the space C^2 of this matrix is Hilbert space. Is it sufficient to generate the inner product: <x,y> = a*ib -1 and obtain the norm by: \begin{equation}...
39. S

### I How to study an ODE in matrix form in a Hilbert space?

Hello, I have derived the matrix form of one ODE, and found a complex matrix, whose phase portrait is a spiral source. The matrix indicates further that the ODE has diffeomorphic flow and requires stringent initial conditions. I have thought about including limits for the matrix, however the...
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### Constructing a 3x3 Linear system question

Homework Statement Construct a 3 × 3 example of a linear system that has 9 different coefficients on the left hand side but rows 2 and 3 become zero in elimination. If the right hand sude of your system is <b1,b2,b3> (Imagine this is a column vector) then how many solutions does your system...
41. M

### Reducing a matrix to echelon form

Homework Statement (i) Reduce the system to echelon form C|d (ii) For k = -12, what are the ranks of C and C|d? Find the solution in vector form if the system is consistent. (iii) Repeat part (b) above for k = −18 Homework Equations Gaussian elimination I used here...
42. S

### I Convert an ODE to matrix form

Hi, I have the following ODE: aY'' + bY' + c = 0 I would like to convert it to a matrix, so to evaluate its eigenvalues and eigenvectors. I have done so for phase.plane system before, however there were two ODEs there. In this case, there is only one, so how does this look like in a matrix...
43. S

### I Can a Hermitian matrix have complex eigenvalues?

Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian? If so, is there any other name to classify it, as it is not...
44. M

### Finding values to make a linear system consistent

Homework Statement Given the following matrix: I need to determine the conditions for b1, b2, and b3 to make the system consistent. In addition, I need to check if the system is consistent when: a) b1 = 1, b2 = 1, b3 = 3 b) b1 = 1, b2 = 0., b3 = -1 c) b1 = 1, b2 = 2, b3 = 3 Homework...
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### Coefficients that make Gaussian elimination impossible?

Homework Statement Given this matrix: I am asked to find values of the coefficient of the second value of the third row that would make it impossible to proceed and make elimination break down. Homework Equations Gaussian elimination methods I used given here...
46. S

### I How can I analyse and classify a matrix?

Hi, I have a matrix of an ODE which yields complex eigenvalues and eigenvectors. It is therefore not Hermitian. How can I further analyse the properties of the matrix in a Hilbert space? The idea is to reveal the properties of stability and instability of the matrix. D_2 and D_1 are the second...
47. ### Using Kirchoff's Voltage Rule to find currents in a system

Homework Statement Then solve these equations for i1-4 Homework Equations V=IR The Attempt at a Solution 80i_1-50i_2-30i_3 = -120 -50i_1+100i_2-10i_3-25i_4 = 0. -30i_1-10i_2+65i_3-20i_4 = 0. -25i_2-20i_3+100i_4 = 0. i_1=-4.18239492 i_2=-2.66455194 i_3=-2.71213323 i_4=-1.20856463...
48. ### I How to find the matrix of the derivative endomorphism?

We have ##B=(1, X, X^2, X^3)## as a base of ##R3 [X]## and we have the endomorphisms ##d/dX## and ##d^2/dX^2## so that: ##d/dX (P) = P'## and ##d^2/dX^2 (P) = P''##. Calculating the matrix in class, the teacher found the following matrix, call it ##A##: \begin{bmatrix} 0 & 1 & 0 & 0...
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### Is the world real?

How we can be sure that we are not living in matrix kind of virtual reality where we even do not have our bodies but all we have is our brain kept in jar of some liquid ? then also how we can be sure that the history as we know it till the last microsecond is totally made up and has been...
50. ### I Bilinear forms

I'm having trouble understanding a step in a proof about bilinear forms Let ## \mathbb{F}:\,\mathbb{R}^{n}\times\mathbb{R}^{n}\to \mathbb{R}## be a bilinear functional. ##x,y\in\mathbb{R}^{n}## ##x=\sum\limits^{n}_{i=0}\,x_{i}e_{i}## ##y=\sum\limits^{n}_{j=0}\;y_{j}e_{j}##...