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), ...
2
votes
4answers
61 views
Diagonalizable matrices in $M_{2\times 2}(\mathbb{F}_2)$
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 ...
3
votes
1answer
69 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, ...
1
vote
2answers
50 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!
1
vote
4answers
22 views
Prove that if $p(A)=0$ where A is a matrix of a linear operator ($A \in L(V)$), $p(\lambda)=0$ if $\lambda \in \sigma(A)$
I think it's all in the title. $p$ is some random polynomial.
I don't know how to approach this one. I've tried taking the roots of $p$, placing them on the diagonal of a new matrix and reasoning that ...
1
vote
1answer
30 views
Some questions on Proof of Structure Theorem
I understand the general idea of the proof of Structure Theorem for finitely generated modules over a principal ideal domain but I found it quite difficult to follow some lines of reasoning in the ...
1
vote
0answers
35 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:
\begin{align*}
P_x &= \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix},
\ P_y = ...
1
vote
2answers
27 views
determinants of 2 matrices with given property
I have two $3\times3$ integer matrices $A$ and $B$ such that $AB=A+B$. I need to find all possibe values of $\det(A-E)$, where $E$ denotes the identity matrix. Any help is appreciated.
4
votes
3answers
514 views
Cayley-Hamilton theorem on square matrices
Can anyone help me by giving the proof of the Cayley-Hamilton theorem? It states that every square matrix $A$ satisfies its own characteristic equation:
$p_{A}(A)=0$.
I could prove it when $A$ has ...
3
votes
3answers
120 views
Let $\alpha$ and $\beta$ be two distinct eigenvalues of $A$ then $ A^3 = \frac{\alpha^3-\beta^3}{\alpha-\beta}A-\alpha\beta(\alpha+\beta)I$?
Let $\alpha$ and $\beta$ be two distinct eigenvalues of a $2\times2$ matrix $A$. Then which of the following statements must be true.
1 - $A^n$ is not a scalar multiple of identity matrix for any ...
0
votes
3answers
50 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, ...
0
votes
2answers
181 views
Elementary Row Operations To Find Inverse Matrix
I have to find the inverse matrix of this matrix that represents a relation. My question is, is it possible to use elementary row operations on a one-zero matrix to find the inverse? I've done it ...
10
votes
2answers
248 views
A matrix w/integer eigenvalues and trigonometric identity
Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated:
Let $n$ be a natural number.
(a) Consider the following Toeplitz/circulant symmetric matrix:
...
1
vote
4answers
636 views
Positive Definite Matrix Determinant
Prove that a positive definite matrix has positive determinant and
positive trace.
In order to be a positive determinant the matrix must be regular and have pivots that are positive which is ...
0
votes
1answer
139 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 ...
3
votes
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 ...
2
votes
1answer
159 views
Diagonalizing a Unitary Matrix
I'm trying to diagonalize the following unitary matrix:
$\frac {1}{\sqrt{5}}\begin{pmatrix} 1 &2 \\ 2i &-i
\end{pmatrix}$
My approach is to find the eigenvalues and eigenvectors in the usual ...
0
votes
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 ...
4
votes
2answers
124 views
Product of matrices; MAPLE giving a strange answer
Either my brain is seriously fried up right now or the computer is wrong.
If I have a matrix $\begin{bmatrix}
4 & -2\\
2 & -1 \\
0 & 0
\end{bmatrix}$ multiply by its transpose ...
2
votes
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
$$ \left( \begin{bmatrix}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & ...
4
votes
3answers
87 views
How to find 3 x 3 matrix inverses
Is there a way of finding the inverse of a $3 \times 3$ matrix without forming an augmented matrix with the identity matrix? Also, is there a quick way of checking that a $3 \times 3$ matrix's ...
3
votes
1answer
69 views
Property of the trace of matrices
Let $A(x,t),B(x,t)$ be matrix-valued functions that are independent of $\xi=x-t$ and satisfy $$A_t-B_x+AB-BA=0$$ where $X_q\equiv \frac{\partial X}{\partial q}$.
Why does it then follow that ...
1
vote
0answers
33 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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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?
6
votes
1answer
63 views
Hall's identity and beyond?
There is a identity, well-known among people that know this sort of things, that is called Hall's identity (or Wagner's identity): for all choices of $2\times 2$ matrices over a fixed field $A$, $B$, ...
2
votes
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 ...
0
votes
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 ...
4
votes
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 ...
1
vote
0answers
27 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 ...
1
vote
1answer
27 views
Composition of systems of equations
Suppose $$2x + 3y = u$$ $$x - 4y = v$$
and further that
$$3u - 5v = c$$ $$2u + 3v = d$$
Express c and d in terms of $x$ and $y$ by matrix multiplication.
It's quite easy by direct substitution but ...
5
votes
1answer
53 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
0
votes
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?
5
votes
4answers
74 views
For $n >1$,let $\displaystyle f(n)$ be the number of $n \times n$ real matrices $A$ such that $A^2+I=0.$
I came across the following problem that says:
For $n >1$,let $\displaystyle f(n)$ be the number of $n \times n$ real matrices $A$ such that $A^2+I=0.$ Then which of the following options is ...
5
votes
6answers
1k views
Sum of all elements in a matrix
The trace is the sum of the elements on the diagonal of a matrix. Is there a similar operation for the sum of all the elements in a matrix?
3
votes
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) \\ ...
0
votes
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 ...
8
votes
2answers
241 views
$AB-BA=I$ having no solutions
The question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I tried block ...
1
vote
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 ...
0
votes
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 ...
16
votes
1answer
709 views
Proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ in Penrose graphical notation
For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation.
Here $\det$ denotes the ...
0
votes
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 ...
0
votes
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 ...
4
votes
1answer
166 views
Are there Taylor series for functions of a matrix?
Say you have a scalar function $f(x,A)$ of a vector $x$ and a matrix $A$. Does there exist a Taylor series of sorts for the matrix $A$? I was thinking naively that this would simply be of the form ...
0
votes
1answer
42 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 ...
1
vote
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 ...
8
votes
3answers
157 views
Find a ternary $4\times 39$ matrix satisfying the conditions below
Can you find a matrix $A_{4\times39}$ with elements from $\{-1,0,1\}$ so that
No column is all zero.
All columns are different.
No column is $-1$ times another column.
Each row consists of $13$ of ...
3
votes
1answer
60 views
Matrices that satisfy $A^2,B^2,C^2$ and commute
Is there a set of matrices that satisfy all of the following constraints?
1) $A^2=0, B^2=0, C^2=0...$ where $A,B,C,D..$ are different matrices.
2) All of them commute.
Edit: 3) $AB \neq 0, AC \neq ...
0
votes
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 ...
1
vote
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 ...
