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), ...
0
votes
0answers
20 views
Inner product space an two orthonormal basis. [duplicate]
Let $V$ be an inner product space. And let $v_1,...,v_n$ and $w_1,...,w_n$ be two orthonormal basis of $V$. How one could show that $[Id]^{v_1,...,v_n}_{w_1,...,w_n}$ is unitary matrix.
3
votes
2answers
25 views
Matrix Equation, Solving for Variables.
I'm going through my exercises, and came across a problem that wasn't covered in our lectures. Here's the question:
$
\begin{align}
\begin{bmatrix}
a-b & b+c\\
3d+c & 2a-4d
\end{bmatrix}
...
0
votes
1answer
21 views
What functions are solution to a homogeneous system of differential equations?
Given a vector $\vec{u} \in \mathbb{R}^n$. For what functions $\psi(t)$ can $\vec{x}(t) = \psi(t)\vec{u}$ be a solution of $\dot{\vec{x}} = A \vec{x}$ for some $n \times n$ matrix $A$?
I'm trying to ...
1
vote
2answers
58 views
Finding invariant factors of finitely generated Abelian group
There is this question that I wasn't sure how to do but somehow got the answers partially correct (maybe).
Suppose that the abelian group $M$ is generated by three elements $x,y,z$ subject to the ...
2
votes
2answers
120 views
Diagonalizability in $\mathbb{R}$ and $\mathbb{C}$
Give an example of a matrix $A\in M_{n\times n}(\mathbb{R})$ that is not diagonalizable, but A is diagonalizable viewed as a matrix over the field of complex numbers $\mathbb{C}.$
-1
votes
0answers
31 views
orthonormal basis linear transformation
A linear transformation
which takes an orthonormal basis into another orthonormal basis
is orthogonal. (T)
I got True for the answer. But can't think of clear explanation of why that is true.
Why it ...
2
votes
1answer
49 views
QR computation only in square matrix A?
I thought the following was true. But the answer is False.
Why so? Could anybody give me some counterexample?
For any matrix A, one can find Q and R such that A = QR
, where Q is an orthogonal matrix
...
0
votes
1answer
39 views
Dimension of vector space and symmetric matrix [duplicate]
Why the following statement is true?
I am so frustrated that I could not have any clue on this problem.
The dimension of the vector space of all symmetric 4 by 4 matrices is 10.
Please help me.
1
vote
1answer
33 views
orthogonal matrix and elementary matrix
Answer is False. But I can't think of the counter example... Could anybody have it?
Let A be an orthogonal 4 x 4 matrix such that $$ Ae_1 = e_2, Ae_2 = e_3, Ae_3 = e_1$$ Then $$Ae_4 = e_4 $$
4
votes
1answer
90 views
General Solution to $\operatorname{Tr} \ln ( I + A)$ where A is complex and symmetric and zero on the diagonal
Are there any useful identities which would help me to find a general formula for
$ \operatorname{Tr} \ln ( I + A ) $
Where I is the identity matrix and A is some N by N complex and symmetric ...
0
votes
0answers
40 views
How to calculate an orthonormal basis for a matrix?
Are there any specific, easy to compute, algorithms to build an orthonormal basis for a matrix in which each column has length one?
1
vote
1answer
29 views
Some questions on invariant factor decomposition of modules
Let $M$ be the $\mathbb{Z}$-module $F/N$ where $F=\mathbb{Z}^2$ and $N=\langle(6,4),(4,8),(4,0)\rangle\le\mathbb{Z}^2$. Consider the homomorphism $\varphi:\mathbb{Z}^3\to\mathbb{Z}^2$ such that the ...
1
vote
4answers
29 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
40 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
2answers
31 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.
1
vote
0answers
43 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 = ...
2
votes
4answers
96 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 ...
0
votes
3answers
59 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, ...
3
votes
3answers
50 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 ...
3
votes
1answer
72 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
73 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!
0
votes
1answer
55 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
3answers
97 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 ...
2
votes
1answer
37 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
1answer
72 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
1answer
39 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
1answer
50 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
1answer
41 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 ...
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 ...
0
votes
1answer
41 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 ...
2
votes
1answer
36 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
$$ \left( \begin{bmatrix}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & ...
1
vote
1answer
27 views
Composition of systems of equations
Suppose $$2x + 3y = u$$ $$x - 4y = v$$
and further that
$$3u - 5v = c$$ $$2u + 3v = d$$
Express c and d in terms of $x$ and $y$ by matrix multiplication.
It's quite easy by direct substitution but ...
1
vote
0answers
28 views
Using a matrix to organise values into groups
Let's say I have a matrix of size 6 x 6.
Six students are 'ranking' six other students (including themselves). If I wanted to organise them into let's say, groups of three without picking and ...
0
votes
0answers
10 views
Can Hessian matrix of probability density function be called density matrix for quantum mechanic
how to calculate density matrix from view of probability for quantum mechanic
Hessian matrix is positive definite, can it be density matrix?
5
votes
5answers
89 views
Symmetric Matrices of $I_{2}$
Find $10$ symmetric matrices $ A = \begin{pmatrix}
a &b \\
c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$
(I'm going to call matrix A the "square root" of $A^{2}$. If this is the incorrect ...
3
votes
2answers
30 views
Matrix multiplication related to complex numbers?
Evaluate and simplify the product
$\begin{bmatrix} r\cos(\alpha) & -r\sin(\alpha) \\ r\sin(\alpha) & r\cos(\alpha)\\ \end{bmatrix}$ $\begin{bmatrix} s\cos(\beta) & -s\sin(\beta) \\ ...
5
votes
1answer
60 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
0
votes
1answer
31 views
Relationship between three matrices
I think this might be an odd question, and a little vague. But here goes.
This is related to coordinate transformations. Three matrices are given: $G_1 , G_2$, and $\Lambda$. $G_1$ and $G_2$ are ...
1
vote
1answer
51 views
Fast way to calculate Eigen of 2x2 matrix using a formula
I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html
Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
0
votes
1answer
160 views
I want help with $4\times 4$ symmetric matrix
I have a $4\times 4$ matrix $$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
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 ...
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
44 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
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
34 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
35 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} $$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues.
My Work:
...
5
votes
1answer
58 views
Is there a name for this given type of matrix?
Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$?
(The motivation for this ...
1
vote
1answer
46 views
Column entries of a matrix sum to zero, so what are the properties?
What kind of properties does a matrix whose column entries sum to zero have?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
