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), ...
1
vote
1answer
15 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 ...
3
votes
0answers
34 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 ...
0
votes
1answer
34 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?
2
votes
2answers
39 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 ...
1
vote
0answers
19 views
Bounds over the eigen-values 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
18 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
0answers
19 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
$$ \left( \begin{bmatrix}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & ...
1
vote
1answer
24 views
Composition of systems of equations
Suppose $$2x + 3y = u$$ $$x - 4y = v$$
and further that
$$3u - 5v = c$$ $$2u + 3v = d$$
Express c and d in terms of $x$ and $y$ by matrix multiplication.
It's quite easy by direct substitution but ...
1
vote
0answers
25 views
Using a matrix to organise values into groups
Let's say I have a matrix of size 6 x 6.
Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
0
votes
0answers
10 views
Can Hessian matrix of probability density function be called density matrix for quantum mechanic
how to calculate density matrix from view of probability for quantum mechanic
Hessian matrix is positive definite, can it be density matrix?
4
votes
4answers
62 views
Symmetric Matrices of $I_{2}$
Find 10 symmetric matrices $ A = \left| \begin{array}{cc}
a & b \\
c & d \\
\end{array} \right|$ such that $A^{2}=I_{2}$
(I'm going to call matrix A the "square root" of $A^{2}$. If this is ...
3
votes
2answers
30 views
Matrix multiplication related to complex numbers?
Evaluate and simplify the product
$\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
5
votes
1answer
51 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
0
votes
1answer
29 views
Relationship between three matrices
I think this might be an odd question, and a little vague. But here goes.
This is related to coordinate transformations. Three matrices are given: $G_1 , G_2$, and $\Lambda$. $G_1$ and $G_2$ are ...
1
vote
1answer
42 views
Fast way to calculate Eigen of 2x2 matrix using a formula
I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html
Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
-1
votes
0answers
72 views
I want help with $4\times 4$ symmetric matrix
I have a $4\times 4$ matrix $$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
43 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 ...
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 ...
0
votes
1answer
40 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 ...
0
votes
1answer
20 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
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 ...
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
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} $$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues.
My Work:
...
5
votes
1answer
56 views
Is there a name for this given type of matrix?
Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$?
(The motivation for this ...
0
votes
1answer
24 views
Column entries of a matrix sum to zero, so what are the properties?
What kind of properties does a matrix whose column entries sum to zero have?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
0
votes
0answers
21 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, ...
1
vote
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 $x(0) = \begin{bmatrix}
2\\
-3
\end{bmatrix}$
0
votes
2answers
49 views
Trace of a matrix
What is the trace of $e^{A}$ where A is a $4 \times4$ matrix
$$\begin{bmatrix}0 & 0 & 0 & t\\
0 & 0 &-t & 0\\
0 & t & 0 & 0\\
-t & 0 & 0 & 0 ...
4
votes
1answer
72 views
Matrix $BA\neq$$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 ...
0
votes
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 ...
4
votes
2answers
78 views
Determinants: A Special Condition
Under what conditions is
$$ \det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n), $$
just curious.
0
votes
0answers
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 ...
0
votes
0answers
18 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 ...
2
votes
3answers
82 views
0
votes
2answers
39 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
votes
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
$$
e^M=\left(\begin{array}{ccc}
e^a & 0 & 0 \\
0 ...
0
votes
1answer
18 views
Why does the non-negative matrix factorization problem non-convex?
Supposing $\mathbf{X}\in\mathbb{R}_+^{m\times n}$, $\mathbf{Y}\in\mathbb{R}_+^{m\times r}$, $\mathbf{W}\in\mathbb{R}_+^{r\times n}$, the non-negative matrix factorization problem is defined as:
...
2
votes
2answers
28 views
Approximation of matrix in 2-norm
The question is the following: Given a matrix $A$ with rank $k$, we are looking for a matrix $B$ of rank $j$, where $j<k$ such that $\|A-B\|_2$ is minimal.
My idea was to choose, if $A=P ...
0
votes
1answer
34 views
Result of multiplying a scaling matrix with a rotation matrix
I don't understand why if you multiply a scaling matrix with rotation matrix that the resulting matrix, when applied to a shape like an ellipse, only gets scaled and does not get rotated.
$$\left( ...
8
votes
1answer
70 views
Find $M$, where $M^7=I$ and $M\neq I$, $M$ has only 0's and 1's.
Find a $3 \times 3 $ matrix $M$ with entries 0 and 1 only such that $M^7=I$ and $M\neq I$.
This was a short question in a recent exam. I tried with permutation matrices but couldn't find $M^{odd}=I$ ...
2
votes
2answers
32 views
Identity of inverse matrix
How can I prove that
$$(C_{N}^{-1} + W_{N})^{-1} = C_{N}(I + W_{N}C_{N})^{-1}$$
I tried to use the Woodbury identity:
$$(A + BD^{-1}C)^{-1} = A^{-1}-A^{-1}B(D+CA^{-1}B)^{-1}CA^{-1}$$
which seems ...
0
votes
2answers
33 views
Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices
Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
-1
votes
3answers
113 views
Evaluation of a specific determinant.
Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$.
$$A_2
\begin{pmatrix} 1& 1\\
1& 2
\end{pmatrix};
A_3 = ...
5
votes
2answers
55 views
Let $A$ be a matrix sized $p\times p$, where $2\le p$. Using recurrence relations, describe $A^k$.
Let $A$ be a matrix sized $p\times p$, Where $2\le p$.
The matrix values in the main diagonal are $0$ and the rest are $1$'s.
Example for $A$ where $p=5$:
$$\begin{bmatrix} 0 & 1 & 1 & 1 ...
0
votes
2answers
31 views
any monic polynomial of degree $n$ whose root is $A$ is the characteristic polynomial of $A$?
Consider any matrix $A$ of order $n\times n.$ How to show that any monic polynomial of degree $n$ whose root is $A$ is the characteristic polynomial of $A?$
I have used the result several times ...
1
vote
1answer
41 views
How to show that $A=B-C$
How to show that for a real symmetric matrix $A,~A$ can be written as $A=B-C$ where $B,C$ are positive definite real symmetric matrices?
Please help me ! I'm clueless.
0
votes
0answers
23 views
Matrix calculus: derivative of product
I'm attempting to find the derivative with respect to $\beta$ of:
$\mbox{RSS}(\beta) = (\mathbf{Y} - \mathbf{X}\beta)^T(\mathbf{Y}-\mathbf{X}\beta),$
where $\mathbf{Y}, \mathbf{X}$ and $\beta$ are ...
1
vote
0answers
22 views
Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?
Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow,
...
