For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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Some questions on Proof of Structure Theorem

I found it quite difficult to follow some lines of reasoning in the proof of Structure Theorem for finitely generated modules over a principal ideal domain. I have spent hours tried to think about it ...
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2answers
14 views

determinants of 2 matrices with given property

I have $2$ square matrices $\;A\;$ AND $\;B\;$ of third order with all the elements integer $\;AB=A+B.\;$ I need to find the possibe values for the determinant $\;|A-E|\;$ , where $\;E\;$ is the ...
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0answers
24 views

What are the vectors $v$ and $w$, given the permutation matrices…

I want to determine the vectors $v$ and $w$, given the following product: $P_x = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$ $P_y = \begin{bmatrix} 0 ...
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3answers
44 views

$M_{2\times 2}(\mathbb{F}_2)$: Diagonizable Matrices

List all diagonalizable $2\times 2$ matrices over the a field $F$ consisting of two elements $0$ and $1$. I want to try and do this using C++, but perhaps this isn't the place to ask. I have an idea ...
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3answers
47 views

Diagonalizable Operators: An Operational Extension

Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator $$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$ on $V$ is also diagonalizable for any scalars $a_1, ...
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3answers
42 views

$ e^{At}$ for $A = B^{-1} \lvert \cdots \rvert B $

For a homework problem, I have to compute $ e^{At}$ for $$ A = B^{-1} \begin{pmatrix} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} B$$ I know how to compute the result ...
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1answer
50 views

Square matrix $||Ax-Ay||\le ||x-y||$

Could you give me an example of a square matrix $A\in \mathcal{M}_{2 \times 2}$ or $\mathcal{M}_{3 \times 3}$ for which we have $||Ax-Ay||\le ||x-y||$, $ \ \ x, y \in \{0, e_1, . . . , e_n\}, \ \ e_1, ...
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34 views

Product of permutation matrices

I want to prove that the product of two permutation matrices is itself a permutation matrix. But I don't know how. Please help!
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1answer
47 views

How to show this matrix is invertible?

Let $f:H \times H \to \mathbb{R}$ be a mapping with $H$ a Hilbert space. Let $A$ be a matrix with entries $a_{ij}=f(b_i, b_j)$ with $$a_{ii}=f(b_i, b_i) \geq C\lVert b_i\rVert_{H}^2.$$ Suppose $b_i ...
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3answers
86 views

How to find 3 x 3 matrix inverses

Is there a way of finding the inverse of a $3 \times 3$ matrix without forming an augmented matrix with the identity matrix? Also, is there a quick way of checking that a $3 \times 3$ matrix's ...
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1answer
34 views

How to frame this set of linear equations?

I have the following set of equations, as an example $2x + 1y + 2z = A$ $0x + 2y + 2z = A$ $1x + 2y + 1z = A$ I assume this can be rewritten as a matrix? How can I check if a solution exists such ...
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1answer
67 views

Property of the trace of matrices

Let $A(x,t),B(x,t)$ be matrix-valued functions that are independent of $\xi=x-t$ and satisfy $$A_t-B_x+AB-BA=0$$ where $X_q\equiv \frac{\partial X}{\partial q}$. Why does it then follow that ...
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1answer
37 views

Special linear transformations

Special linear transformations are matrices with determinant equal to 1. What additional properties do such transformations have compared to "regular" linear transformations?
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1answer
48 views

If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?

We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm. Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 ...
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0answers
32 views

Proof is needed for a lower bound of the maximal eigen-value of a non-negative, irreducible, integer matrix

$A$ is a non-negative, integer, irreducible, $m$ by $m$ matrix. It is well known (Perron-Frobenius) that $A$ has a positive eigen value (denote it by $\lambda$) with a positive eigen vector ($x$). It ...
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1answer
32 views

need help solving - system of equations

i was writing a model paper for a olympiad when i encountered this question: i thought of using cramer's rule or just proceed with matrix inversion method but i am stuck trying to figure it out. is ...
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1answer
22 views

Decreasing the computational speed of Gaussian elimination of a complex linear system in a special case.

The solution of the complex linear system $Ax = b$ of $n$ equations can be computed using Gaussian elimination with $O(n^3)$ complex multiplications. However, how can we show that if ...
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1answer
29 views

If $\omega$ is a complex cube root of unity, show that the following equals null matrix.

If $\omega$ is a complex cube root of unity, show that $$ \left( \begin{bmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & ...
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1answer
27 views

Composition of systems of equations

Suppose $$2x + 3y = u$$ $$x - 4y = v$$ and further that $$3u - 5v = c$$ $$2u + 3v = d$$ Express c and d in terms of $x$ and $y$ by matrix multiplication. It's quite easy by direct substitution but ...
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26 views

Using a matrix to organise values into groups

Let's say I have a matrix of size 6 x 6. Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
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0answers
10 views

Can Hessian matrix of probability density function be called density matrix for quantum mechanic

how to calculate density matrix from view of probability for quantum mechanic Hessian matrix is positive definite, can it be density matrix?
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4answers
68 views

Symmetric Matrices of $I_{2}$

Find 10 symmetric matrices $ A = \left| \begin{array}{cc} a & b \\ c & d \\ \end{array} \right|$ such that $A^{2}=I_{2}$ (I'm going to call matrix A the "square root" of $A^{2}$. If this is ...
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2answers
30 views

Matrix multiplication related to complex numbers?

Evaluate and simplify the product $\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
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1answer
53 views

Having trouble using eigenvectors to solve differential equations

The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix} 5 & 4 \\ -1 & 1\\ \end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix}$$ I went ...
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1answer
29 views

Relationship between three matrices

I think this might be an odd question, and a little vague. But here goes. This is related to coordinate transformations. Three matrices are given: $G_1 , G_2$, and $\Lambda$. $G_1$ and $G_2$ are ...
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1answer
44 views

Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
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1answer
131 views

I want help with $4\times 4$ symmetric matrix

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
0
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1answer
44 views

How the inverse of this matrix be found?

How can the inverse of matrix $A = \left( \begin{smallmatrix} 6&5\\5&4 \end{smallmatrix} \right)$ be $A^{-1} = \left( \begin{smallmatrix} -4&5\\ 5&-6 \end{smallmatrix} \right)$ where ...
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0answers
7 views

How to decompose a matrix into tensor product of Hermitian matrix

How to decompose a matrix into tensor product of Hermitian matrix is there a algorithm to do this? or pseudo code? bonus: is it possible to decompose a matrix into a linear combination of tensor ...
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0answers
15 views

How to calculate orthogonal projection of one dimension vector

refer to http://mathoverflow.net/questions/60185/linear-combination-of-orthogonal-projection-matrices if use one dimension vector to calculate orthnormal basis by Gram-Schmidt algorithm. then how to ...
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1answer
41 views

How to generate a N*D random matrix with columns of unit length?

Is it possible to generate a N*D random matrix with columns of unit length? If not, I also think it is possible of generating a N*D random matrix and, after that, normalizing it in order to have ...
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1answer
21 views

Matrix Multiplication with Transponse

When you multiply a matrix M by its transpose, what exactly does this product represent, what do each value in the cell represent? I see that a lot of these examples, when a document term matrix ...
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1answer
32 views

Computing Resultant

The resultant of two polynomials is defined as the determinant of the Sylvester matrix. If the polynomials are of degree $n$ and $m$, than the Sylvester matrix will be of dimension $(m+n)\times ...
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1answer
17 views

What is the meaning of 'columns have unit lengths'

What is the meaning of this? In random projection, the original d-dimensional data is projected to a k-dimensional (k << d) subspace through the origin, using a random k × d matrix R ...
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3answers
34 views

Eigenvector Proof $(I+A)^{-1}$.

Show that the eigenvectors of the $n \times n$ matrix A are also eigenvectors of the matrix $$M = (I+A)^{-1} $$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues. My Work: ...
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1answer
57 views

Is there a name for this given type of matrix?

Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$? (The motivation for this ...
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1answer
26 views

Column entries of a matrix sum to zero, so what are the properties?

What kind of properties does a matrix whose column entries sum to zero have? $$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & ...
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0answers
23 views

Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector

Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries. The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has all coordinates, ...
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1answer
38 views

Solve a System with Variable

Given these matrices, how does one find two real solutions? $dx/dt$ = $\begin{bmatrix} 3 & -5\\ 5 & 3 \end{bmatrix}x$ with $x(0) = \begin{bmatrix} 2\\ -3 \end{bmatrix}$
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2answers
49 views

Trace of a matrix

What is the trace of $e^{A}$ where A is a $4 \times4$ matrix $$\begin{bmatrix}0 & 0 & 0 & t\\ 0 & 0 &-t & 0\\ 0 & t & 0 & 0\\ -t & 0 & 0 & 0 ...
4
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1answer
75 views

Matrix $BA\neq$$I_{3}$

If $\text{A}$ is a $2\times3$ matrix and $\text{B}$ is a $3\times2$ matrix, prove that $\text{BA}=I_{3}$ is impossible. So I've been thinking about this, and so far I'm thinking that a homogenous ...
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0answers
18 views

Notation for Hadamard division

What is a reasonable notation for Hadamard division of two matrices? Several forum threads point to $\oslash$ as a possibility, but it feels "forced", for lack of a better word (I might go with a ...
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2answers
82 views

Determinants: A Special Condition

Under what conditions is $$ \det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n), $$ just curious.
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15 views

The definition of “projector” when it is not a linear system.

From any linear algebra book, projection is defined as the best solution for |y-Ax| under L2 norm. My problme is, if I don't model a system as y=Ax , but instead using another function y=f(x), if I ...
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0answers
19 views

Condition number of a function

I would like to find the Condition number of a function (f(x)) with one variable (x) and several parameters. which can be calculated by: $$ c(x) = ||x||*||f'(x)||/||f(x)||$$ Here if my function is ...
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3answers
83 views

Why is $\det⁡(-A)=(-1)^n\det(A)$? [closed]

Why is $\det⁡(-A)=(-1)^n\det(A)$?
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2answers
40 views

Which of the following are subspaces of $M$?

Let $M$ be a vector space of all $3\times 3$ real matrices and let $$A=\begin{pmatrix}2&3&1\\0&2&0\\0&0&3\end{pmatrix}.$$ Which of the followings are subspaces of $M?$ ...
2
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2answers
65 views

For a diagonal matrix $M$, what is $e^M$?

For a diagonal matrix $$ M=\left(\begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right) $$ show that $$ e^M=\left(\begin{array}{ccc} e^a & 0 & 0 \\ 0 ...
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1answer
18 views

Why does the non-negative matrix factorization problem non-convex?

Supposing $\mathbf{X}\in\mathbb{R}_+^{m\times n}$, $\mathbf{Y}\in\mathbb{R}_+^{m\times r}$, $\mathbf{W}\in\mathbb{R}_+^{r\times n}$, the non-negative matrix factorization problem is defined as: ...
2
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2answers
29 views

Approximation of matrix in 2-norm

The question is the following: Given a matrix $A$ with rank $k$, we are looking for a matrix $B$ of rank $j$, where $j<k$ such that $\|A-B\|_2$ is minimal. My idea was to choose, if $A=P ...

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