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
3answers
23 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 ...
4
votes
1answer
31 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
7 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
70 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
28 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
26 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
238 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
41 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 ...
-2
votes
0answers
64 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
votes
1answer
41 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
708 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
6 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
164 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
34 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
31 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
4answers
145 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 ...
1
vote
1answer
32 views
Truncated exponential map from $\mathfrak{gl}_n$ to $GL_n$
Let $k$ be a field of characteristic $p>0$. If $A$ is a nilpotent matrix in $\mathfrak{gl}_n(k)$, with $p>n$, then we can define the unipotent matrix:
...
0
votes
1answer
19 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 ...
1
vote
3answers
33 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:
...
5
votes
1answer
56 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 ...
1
vote
0answers
68 views
Are the signs of these eigenvalues from this Hermitian matrix equal to the Möbius function?
I am partly repeating myself here. Are the signs of these eigenvalues from this Hermitian matrix "c" equal to the Möbius function? Eigen99 in the Mathematica code is the list of eigenvalues for a ...
5
votes
1answer
511 views
Is there a unique solution for this quadratic matrix equation?
The quadratic matrix equation I've been looking at lately:
$$ Q_{r,r}=A_{r,r}X_{r,r}^2+B_{r,r}X_{r,r}+C_{r,r}=0_{r,r} $$
Note that $A, B, C,$ and $X$ are $r \times r$ matrices. $A$ contains known ...
2
votes
2answers
402 views
given $y = a + bx + cx^2$ fits three given points, find and solve the matrix equation for the unknowns $a,b$, and $c$
Given $y = a + bx + cx^2$ fits three given points, find and solve the matrix equation for the unknowns $a$, $b$, and $c$.
the equation fits the points $(1,0), (-1, -4),$ and $(2, 11)$
I really ...
2
votes
2answers
146 views
Determine all $2\times2$ matrices $A$ such that $A^2=0$.
This is from Lang's introduction to Linear Algebra page no 61.
Determine all $2\times 2$ matrices $A$ such that $A^2 = 0$.
Let $A=\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$
...
1
vote
2answers
169 views
Matrices that satisfy $A^2=0$ and anticommutation.
Suppose that there is a set of matrices.
1) $A^2 =0, C^2=0, E^2=0 .....$ and $AB+BA=0$, $CD+DC=0$... where $A,B,C..$ are matrices.
2) Matrices in the set either anticommute or commute.
3) $AD+BC ...
1
vote
1answer
164 views
Matrices that satisfy $AB = 0$ and $A^2 = B^2$
I want to make a set of matrices that satisfies all the following:
1) $A^2 = B^2$, $C^2 =D^2$..... where $A,B,C,D...$ are matrices
2) $AB = 0$, $CD = 0$.....
3) All matrices in the set commute.
4) ...
0
votes
5answers
129 views
Satisfying $AB=0$, commute and $A^2+B^2=0$
I heard somewhere that $AB=0$ is related to $A^2+B^2$.
So, does $AB=0$ result in $A^2+B^2 =0$?
Or if it doesn't, which matrices would satisfy $AB=0$ while $A^2+B^2=0$
Edit: right. stupid me. So, ...
1
vote
1answer
215 views
Quadratic Matrix Equation
Consider real diagonal (known) matrices $A$, $B$, and $C$, and a comforming matrix $\Pi$.
$C + \Pi B + \Pi A \Pi = 0$
I have been trying to solve this system using elementary algebra. I have two ...
3
votes
1answer
108 views
Solution to a Matrix equation
Is there a general solution to the following matrix equation.
$A - BAB^T = C$
where B is known but can be any non-symmetric square matrix, C is known and invertible, all are n by n matrices. Is there ...
6
votes
1answer
67 views
small rank solution of a matrix equation
Consider the matrix equation
$$AX-XA = R$$
where $A$ and $R$ are given square matrices such that $\operatorname{rank}(R)=r$. How to establish conditions (necessary, sufficient, or both) on $A$ and ...
1
vote
2answers
80 views
Solving matrix equation $A^n x = b$ [duplicate]
Possible Duplicate:
Solve $A^nx=b$ for an idempotent matrix
It is given that:
$$ A = \begin{bmatrix}
2 & 3 & -4 \\ 0 & 1 & 0 \\ 1/2 & 3/2 & -1
\end{bmatrix} $$
and
...
7
votes
3answers
193 views
Solving matrix equation $XA=AY$ with known $X$ and $Y$
I am having problem in solving set of matrices multiplication.
There are three matrices $A,X$ and $Y$, all are non-singular $2\times 2$ matrices. Where matrix $X$ and $Y$ are known and $A$ is unknown. ...
1
vote
1answer
180 views
How to solve X*A=C matrix equation where two (X and A) matrices are unknown?
I have a spectroscopy problem that boils down to a matrix equation where X*A=C. I take N observations each consisting of 3 detector readings and my detectors suffer from some amount of cross-talk ...
3
votes
2answers
81 views
Linear Algebra matrix $Ax=b$ true or false nullspace
$Ax=b$
$m$ number of Rows
$n$ number of columns
true or false
A) If $n > m$, given any $b$ you can always solve $Ax=b$.
The answer: False. Counterexample: A is the zero matrix.
We have ...
8
votes
2answers
117 views
Solving matrix equations of the form $X = AXA^T + C$
I'm trying to solve this matrix equation:
$$X = AXA^T + C$$
In particular,
$$
X = \begin{bmatrix} 1.5 & 1 \\ -0.7 & 0 \end{bmatrix}
X
\begin{bmatrix} 1.5 & -0.7 \\ 1 & 0 ...
4
votes
1answer
697 views
Solving matrix equations of the form $XA = XB$
I am trying to solve the matrix equation of the form $XA = XB$. $A$, $B$ and the solution sought $X$ are $4 \times 4$ homegeneous matrices which are composed of a rotation matrix and translation ...
-1
votes
3answers
122 views
Need Help with Proof please help! [closed]
Im not sure really how to do this, will someone please help.
Given that $A$ and $B$ are $n\times n$ matrices over field $F$, prove that the solution set $S$ to the matrix equation $XA+BX=0$ is a ...
1
vote
1answer
29 views
Rearragning matrix equation to find a matrix with multiple occurences
Given the equation:
$T+TD+TR=Y$
I need to solve this for $T.$ I have dealt with more basic matrix equations and using the inverse to re-arrange formulas, but since there is an addition of the ...
11
votes
4answers
148 views
Finding all matrices $B$ such that $AB=BA$ for a fixed matrix $A$
Let
$$ A=\begin{pmatrix}
1 & 0 & 0 \\
0& 1 & 0 \\
3 & 1 & 2
\end{pmatrix}
$$
Find all matrices $B$ such that $AB=BA$.
Attempt at solution: I can show that $A$ is ...
1
vote
1answer
107 views
Solution of matrix equation or matrix inequality
When there exist a positive definite solution $S$ of the following matrix equation or matrix inequality:
$
SA^{T}+AS+\alpha S-\beta BB^{T}=0
$,
that is, what condition on $A,B, \alpha, \beta$ can ...
1
vote
1answer
102 views
Solution of a matrix equation
Let $A, B$ be given $n\times n$ matrices with positive entries. Does the matrix equation $XAY=B$ always has a solution $X, Y$ with nonnegative entries?
I tried to use Kronecker product to transform ...
0
votes
1answer
117 views
Matrix equation manipulation
I'm trying to manipulate a matrix equation I've got. Here's what it looks like:
$R_\mathrm{app} = U \cdot F^T$
where $R_\mathrm{app}, U, F$ are matrices. $R_\mathrm{app}$ is a size of $u \times a$, ...
4
votes
0answers
243 views
How to solve matrix equation $AX+XB=C$ for $X$
How to solve matrix equation $AX+XB=C$ for $X$?
It seems not too difficult, I tried many times but fail.
I'm an adult student... I am now vexed about Gilbert Strang - An Introduction to Linear ...
0
votes
2answers
153 views
solving matrix equation
I want to solve the following matrix equation. Could anyone give me a hand? Thanks.
Given an $n \times n$ matrix $\mathbf A$ (diagonally dominant), I need to solve an $n \times n$ symmetric matrix ...
2
votes
1answer
448 views
Solving a matrix equation $AX=XB$ in a CAS
I have the following computational problem. Let $N$ be a positive integer and $A\in \mathbb{C}^{2N\times 2N}$, $X\in \mathbb{C}^{2N\times 4}$ and $B\in \mathbb{C}^{4\times 4}$. I want to solve the ...
1
vote
1answer
96 views
Matrix equation $(Ax - y)^T \cdot A = 0_n^T$ solve for $x$
I have the following matrix equation
$$(A x - y)^T \cdot A = 0_n^T$$
Whereby $x \in \mathbb{R}^n$, $y \in \mathbb{R}^d$, $A \in \mathbb{R}^{d \times n}$. A is not invertible, but $A^TA$ is ...
