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
36 views
Eigenvalues and eigenvectors of AB and BA, proof.
$A$ is an $n \times k$ matrix and $B$ is an $k \times n$ matrix.
If $v_1, ..., v_l$ are linearly independent eigenvectors of $BA$ corresponding to a single nonzero eigenvalue $c$, then $Av_1, ..., ...
0
votes
1answer
24 views
Is there a dot product with which the following linear operator becomes Hermitian
Given the linear operator
$A \in L(M_2(\mathbb{C}))$
$A \begin{bmatrix}a & b \\ c & d \end{bmatrix}=\begin{bmatrix}a-b & -a+b \\ d & -c \end{bmatrix}$
Is there a dot product where ...
0
votes
1answer
28 views
find matrix such that $ Ax=(1,1,1)^t$ has exactly three distinct solutions
Does there exist a matrix $3\times 3$ order such that $ Ax=(1,1,1)^t$ has exactly three distinct solutions? If so, find $A$.
I have no idea in this question please help.
0
votes
0answers
28 views
to find the eigenvalues and eigenvectors from linear transformation
Find the eigenvalues and eigenvectors of the linear transformation $T$:$R^3\to R^3$ defined by $T(x_1,x_2,x_3)=(x_1,x_2,x_3)$?
Please tell me how to find the matrix and then I can find the ...
0
votes
0answers
8 views
Minimum Distance of a Matrix gen
Given the https://kattis.csc.kth.se/problem?id=codes problem, I've get to solve in this way.
I multiply the original matrix by each of the $2^k$ combinations possible and save the minimum Hamming ...
0
votes
1answer
16 views
Schur decomposition of an $n-$by$-n$ matrix
$(\lambda, x)$ is a simple (with multiplicity 1) eigenpair of $A\in \mathbb C_n$ with $x^Hx=1$, $H$ denotes Hermitian.
Use Schur decomposition to show that there exists a nonsingular matrix $(x\ \ ...
2
votes
5answers
74 views
Finding the determinant of $2A+A^{-1}-I$ given the eigenvalues of $A$
Let $A$ be a $2\times 2$ matrix whose eigenvalues are $1$ and $-1$. Find the determinant of $S=2A+A^{-1}-I$.
Here I don't know how to find $A$ if eigenvectors are not given. If eigenvectors are ...
3
votes
1answer
33 views
Solubility From Row Echelon Form
Here is the question I am attempting to solve
Determine which values of $k$, if any, will give: a) A unique solution, b) No solution, c) Infinitely many solutions to the system of equations.
...
2
votes
1answer
35 views
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$
$A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$, consider the linear map $T:M_2(\mathbb{R})\to M_2(\mathbb{R}):=B\to AB$ Then which of the following are true?
$T$ is ...
2
votes
1answer
58 views
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is
Diagonalizable
Positive semidefinite
$0,3$ are only eigenvalues of $J$
Is positive definite
$J$ has minimal polynomial $x(x-3)=0$ so 1, ...
3
votes
2answers
34 views
Given a triangle with points in $\mathbb{R}^3$, find the coordinates of a point perpendicular to a side
Consider the triangle ABC in $\mathbb{R}^3$ formed by the point $A(3,2,1)$, $B(4,4,2)$, $C(6,1,0)$.
Find the coordinates of the point $D$ on $BC$ such that $AD$ is perpendicular to $BC$.
I believe ...
1
vote
1answer
19 views
Computing the expected value of a matrix?
This question is about finding a covariance matrix and I wasn't sure about the final step.
Given a standard $d$-dimensional normal RVec $X=(X_1,\ldots,X_d)$ has i.i.d components $X_j\sim N(0,1)$. ...
1
vote
0answers
19 views
Threshold dense adjacency matrix
I have a dense, adjacency matrix (square, symmetric) representing a graph. I want to threshold that graph so that it only contains the largest weights (cells in the matrix), but is still fully ...
-3
votes
1answer
48 views
Fantastic Determinant (all $b$ plus multiple of $I$) [duplicate]
$$f(a,b)=\operatorname{det}~\begin{pmatrix} a & b & b & \cdots & b \\ b & a & b &\cdots & b\\ b & b & a &\cdots & b\\ \vdots & \vdots & \vdots ...
0
votes
1answer
49 views
Prove: symmetric positive matrix multiplied by skew symmetric matrix equals 0
My teacher gave me this task as preparation for the exam but I'm stuck and not sure if it's true anymore.
1
vote
0answers
27 views
Inverse function of product of exponential matrices
I am looking for the value of $\mathbf{X}$ in a function of the type
\begin{align}
(\mathbf{X}-\mathbf{A})e^{\mathbf{X}}e^{-\mathbf{A}} = \mathbf{B}
\end{align}
where ...
1
vote
1answer
27 views
Symmetric Matrices Using Pythagorean Triples
Find symmetric matrices A =$\begin{pmatrix} a &b \\ c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$.
Alright, so I've posed this problem earlier but my question is in regard to this ...
7
votes
1answer
31 views
Trace of a differential operator
Given the differential operator:
$$A=\exp(-\beta H)$$
where $$H=\frac{1}{2}\left( -\frac{d^2}{dx^2}+x^2 \right)$$
and $\beta\gt 0$
How can I get the trace of this operator?
Thanks in advance.
0
votes
0answers
18 views
Spectral Properties of Concatenated Matrices
I am wondering if anyone is aware any resource on the internet that systematically studies the spectral properties of submatrices/ concatenated matrices.
I am interested in proving/ disproving the ...
1
vote
4answers
86 views
A problem on matrices: Find the value of $k$
If $
\begin{bmatrix}
\cos \frac{2 \pi}{7} & -\sin \frac{2 \pi}{7} \\
\sin \frac{2 \pi}{7} & \cos \frac{2 \pi}{7} \\
\end{bmatrix}^k
=
...
1
vote
4answers
96 views
A problem on matrices : Powers of a matrix
If $ A=
\begin{bmatrix}
i & 0 \\
0 & i \\
\end{bmatrix}
, n \in \mathbb N$, then $A^{4n}$ equals?
I guessed the answer as $ A^{4n}=
\begin{bmatrix}
...
0
votes
1answer
27 views
A problem on matrices : Sum of elements of skew-matrix
If $A=[a_{ij}]$ is a skew-symmetric matrix, then write the value of $$ \sum_i \sum_j a_{ij}$$
My doubt is that what is the meaning of $ \sum_i \sum_j ?$ Is it the same as $\sum_{ij}?$
Please ...
1
vote
2answers
17 views
Determinant of product of symplectic matrices
In optical ray tracing it's possible to use symplectic matrices. I have a problem with them.
If a matrix $M$ is symplectic, this means that for $M$ the following equation hols:
$$M^T\Omega M=\Omega$$
...
0
votes
0answers
9 views
Orthogonalising the standard finite element hat function basis - Mass matrix
If one wants to find the $L^2$ projection of a function f the standard finite element space $V_n$ spanned by basis functions $\{\varphi_i\}_{i=1}^N$, then you solve
$A\alpha=\beta$ where ...
1
vote
1answer
25 views
Finding a matrix with the following property
I have one $n \times n$ symmetric matrix $B$. Let $p$ be a scalar, I want to multiply the diagonal elements of $B$ by $p$. Let now $C$ denote the resultant matrix of the process described. Is there ...
1
vote
2answers
71 views
Upper and Lower Triangular Matrices
Given the matrix A=$ \left( \begin{array}{ccc}
1 & 2 & 3 & 4 \\
5 & 6 & 7 & 8\\
1 & -1 & 2 & 3 \\
2 & 1 & 1 &2\end{array} \right) $, write it in the ...
2
votes
1answer
24 views
Minimal polynomial matrix
I want to show that $ x^n-1$ is the minimal polynomial of the permutation matrix $P:=(e_2,e_3,....,e_n,e_1)$ where $e_i$ is the i-th unit vector written as a column vector.
And now I have to show ...
3
votes
0answers
44 views
Eigenvalues of a tridiagonal trigonometric matrix
Let $A$ be the diagonal matrix w/alternating in sign diagonal entries:
$$ A =
\begin{pmatrix}
\pm \tan(\frac{\pi}{2n+1}) & 0 & 0 & \ldots & 0 \\
0 & \mp ...
2
votes
1answer
35 views
inequality applied to Matrix possible?
My question is this : when is it possible to apply (if at all) a polinomial inequality like this little inequality conjecture ,for example, to a $n\times n$ Matrix $A$ (change the variable $x$ with ...
3
votes
1answer
23 views
Square of sum of matrices
I'm trying to follow these lecture notes on Linear Discriminant Analysis (LDA) but I can't seem to figure out how the author gets from:
$$ \Sigma_{x\epsilon\omega_{i}} (w^{T}x - w^{T}\mu_{i})^2$$
to
...
2
votes
1answer
31 views
Matrices manipulation
I am having difficulty with the following question
I have to determine if the following claim is true or not.
If it is true I have to proof it else I need to give an example
I believe it is not ...
0
votes
1answer
30 views
Can the second term of the Schur complement of a symmetric matrix be undefined?
Given the next symmetric matrix conformably partitioned
$$\begin{bmatrix}
A &B \\ B^T &C
\end{bmatrix}$$
I know that $A$ and $C$ are positive definite matrices.
The Schur complement is ...
2
votes
2answers
23 views
Calculating the centralizer of a matrix in a general linear group.
Let $G = GL(3,\mathbb{R})$ be the general linear group over the reals , of order $3$ , and let
$A\in G$ be :
$$
A=\begin{pmatrix}
-1 & 0 & 0 \\\
0 & 1 & 0 \\\
0 & 0 & 2
...
1
vote
1answer
20 views
How to show that every complex matrix with orthonormal columns can be supplemented into an unitary matrix?
Show that every matrix $A \in M_{n,k}(\mathbb{C})$ whose columns are orthonormal vectors in $M_{n1}(\mathbb{C})$ can be supplemented with additional n-k columns to an unitary matrix $U \in ...
2
votes
1answer
34 views
Why $\operatorname{rank}(A^* A)=\operatorname{rank}(A)$ is equivalent to $A^* Ax=0$ if and only if $Ax=0$?
Let $A \in M_{m\times n}(F)$ and $x \in F^n$.
$A^*$ is the adjoint of $A$.
Why is $\operatorname{rank}(A^* A)=\operatorname{rank}(A)$ equivalent to $A^* Ax=0$ if and only if $Ax=0$?
0
votes
1answer
18 views
Matrix inverses over finite fields with composite moduli
I know that over a field $F$, a matrix is invertible if and only if its determinant is nonzero. And I understand why this is true, at least in the case where the field is $\mathbb{R}$.
But I do not ...
2
votes
1answer
70 views
How prove that $\;(1-Tr(A))^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4\;\;?$
Let $A=\begin{bmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}
\end{bmatrix}$ be an orthogonal matrix with $a_{i,j}\in \mathbb R$, where $\det(A)=1$
...
1
vote
0answers
12 views
root mean square deviation value using kabsch
i've got myself to a problem, where i am implementing Kabsch algorithm to calculate root-mean-square-deviation.
I'm using two matrices to get a rotation and translation matrices. (Java implementation ...
2
votes
0answers
41 views
Proof of Sum, Difference, Scalar Multiple of Diagonal Matrices
Assumming A and B are diagonal matrices of the same size, please prove that the following are diagonal matrices as well.
a) $A+B$
b) $A-B$
c) $kA$ , for a scalar $k$
It's not homework- just a ...
1
vote
1answer
36 views
$(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices
Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
1
vote
1answer
28 views
Why is the square of a normal matrix as well a normal matrix?
Or is it? Be $A$ a normal matrix and my question is if $A^2$ is as well a normal matrix?
1
vote
0answers
31 views
Basis of kernel and image of a linear transformation - verification
The transformation matrix I found is: $$\begin{pmatrix} 1 & -1 \\ 1 & 1 \\ 0 & 0\end{pmatrix}$$
Is this how a basis for $\ker$ and $\mathrm{im}$ is calculated?
$$\begin{pmatrix} 1 & ...
1
vote
0answers
22 views
How to compress a linear operator and have the lossless composition property.
Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
1
vote
1answer
35 views
Element by element formulae for 3x3 matrix inversion
Given a 3 x 3 matrix:
$$
A= \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
$$
Can $A^{-1}$ be shown as as a 3x3 ...
2
votes
1answer
36 views
A matrix problem :)
If $l_i,m_i,n_i$ ; $i=1,2,3$ denote the direction cosines of three mutually perpendicular vectors in space, provided that $AA^T=I$ ,where
$$A=\begin{bmatrix}
l_1 & m_1 & n_1 \\
...
1
vote
3answers
36 views
A problem on square matrices
If $B,C$ are $n$ rowed square matrices and if $A=B+C, BC=CB, C^2=O$, then show that for every $n \in \mathbb N$, $$A^{n+1}=B^n(B+(n+1)C)$$
I tried to prove it using mathematical induction. But I ...
1
vote
0answers
11 views
Estimate for a rigid transform given a set of noisy measurements
I have a set of rigid transforms $\in \mathbb{R}^{4x4}$, where each transform is an approximation to some unknown, "correct" transform. I'm looking for an algorithm to estimate the correct transform ...
0
votes
0answers
25 views
Quadratic form of block matrix
If one has a block matrix $\tilde A = \left[ {\begin{array}{*{20}{c}}
D&{{0_{n \times n}}}\\
{{0_{n \times n}}}&{{0_{n \times n}}}
\end{array}} \right]$ where $D\in {R^{n \times n}}$ is a ...
1
vote
1answer
34 views
Multiplicity of an eigenvalue is equal to $\dim V_{\lambda}$
I am trying to prove that multiplicity of an eigenvaliue $\lambda$ = $\dim V_{\lambda}$ and I have problems with this inequality:
$\dim V_{\lambda} \le $ multiplicity $\lambda$.
I know that ...
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.
