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), ...
3
votes
0answers
28 views
Definition and some elementary properties of the “vector turn map”
This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
0
votes
1answer
47 views
Eigenvectors of a Matrix Homework
Find eigenvectors of:
\begin{pmatrix}4 & 2 \\ 2 & 3 \end{pmatrix}
$$\det(A-\lambda I) = \lambda^2-7 \lambda+8=0 \iff \lambda_1=\frac{7+\sqrt{17}}{2} \ \lor \ \lambda_2= \frac{7-\sqrt{17}}{2}$$
...
8
votes
5answers
159 views
Matrices with eigenvalues 0 and 1
How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1?
My attempt:
I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
3
votes
0answers
27 views
If the matrix is positive definite, then its similar matrix is also positive definite?
If $A$ is positive definite and $B$ is similar to $A$.
Can we say that $B$ is also positive definite?
I guess it is true since two matrices have same eigenvalues, and if $\sigma(A) > 0$, and so is ...
1
vote
2answers
73 views
Is $\operatorname{GL}(n, \mathbb{R})$ with multiplication a group?
I am looking at an exercise that saying $GL(n,\mathbb{R})$ with multiplication, in other words the nxn matrices with real entries together with multiplication is a group. I wonder the following: do ...
3
votes
1answer
79 views
Skew symmetric matrix decomposes
I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $ R $ such that $ R^T A R = M$, ...
0
votes
0answers
34 views
Vandermonde matrix and polynomials
Question attached as image, deals with polynomials of order N and determinant of
Vanderbilt matrix.
1
vote
0answers
9 views
Two dimensional rotation of a tensor component UV
I have a quantity $[UV]$ where the $[\hspace{5pt}]$ represents ensemble averaging and $U, V$ are the velocity components in $X$ and $Y$ directions. In a new reference frame (say: $x, y$), which is ...
1
vote
0answers
12 views
Eigen values of L2 projection Matrix
If I have an arbitrary set of $L^2$ functions $\{\phi_i\}$, then want to find the projection onto the subspace of $L^2$ generated by the basis, i.e $span\{\phi_i\}$, I believe I just need to solve
...
0
votes
1answer
50 views
Linear algebra determinants
I have tried to solve this problem but I don't have an idea how to begin, any hints?
For any vector $x$ in $\mathbb{R}^n$ let $(x,x) =\sum\limits_{i=1}^n x_i^2 $ . Let $A$ be a matrix of size $n ...
8
votes
4answers
153 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
52 views
Counting 0-1 matrices up to symmetry
I'm interested in counting the number of n×n 0-1 matrices with a given number of 1s up to rotation and reflection. What is the best way to do this if n is not too small?
For example, consider ...
0
votes
1answer
26 views
Tests for positive definiteness for nonsymmetric matrices
Do tests for positive definiteness for nonsymmetric matrices exist? More specifically I am working with bidiagonal upper/lower triangular matrices with positive eigenvalues and I need to check to see ...
0
votes
2answers
41 views
Proving technique for proving $BB^T$ is positive semidefinite for rectangular $B$
Suppose $B$ is an $m \times n$ matrix. Prove that $BB^T$ is positive semidefinite.
Can someone give a fairly good proof?
Inputs are greatly appreciated. The question is listed above.
3
votes
3answers
118 views
Is this map a known one?
Let $A$ be a $2\times2$ real matrix, then define $f:S^1\to S^1$ by $f(\phi)=\arctan(A\cdot(\cos \phi,\sin \phi))$.
This can be viewed as a discrete dynamical system on $\mathbb{S}^1$ and I am trying ...
1
vote
1answer
68 views
How to calculate the determinant of a matrix with …
How to calculate the determinant using Laplace?
$$
\det \begin{bmatrix}
-t & 0 & 0 & \dots & 0 & a_1 \\[0.3em]
a_2 & -t & 0 & \dots & 0 ...
0
votes
2answers
83 views
+50
$X∈M_{m×1}(F)$ and $Y∈M_{1×n}(F)$: A Range Dimension Implication
Let $A \in M_{m \times n}(F)$. To prove that $\operatorname{rank}(A) \le 1$ if and only if there exist $X \in M_{m \times 1}(F)$ and $Y \in M_{1 \times n}(F)$ such that $A=XY$ where must I start?
1
vote
1answer
32 views
$\operatorname{rank}(A\in M_{m\times n}(F)) =m \implies \exists~B\in M_{n\times m}(F)$ s.t. $AB=I_m$
Let $A ∈ M_{m×n}(F)$ be a matrix with $\operatorname{rank}(A) = m$. I just need some help showing that there exists a matrix $B ∈ M_{n×m}(F)$ such that $AB = I_m$.
1
vote
2answers
44 views
Proving something is a square matrix
I don't want the solution. Please don't post the full solution. I just need a starting clue on how to do this.
Suppose $A$ and $B$ are matrices such that $AB$ and $BA$ are defined.
a) Show that ...
0
votes
1answer
29 views
has a diagonalizable matrix jordan normal form or not?
i would like to ask if a matrix that is diagonalizable then that means that i hasn't Jordan normal form .
if that's not the case then please tell me some ways to check if a matrix has jordan normal ...
0
votes
3answers
68 views
Finding normalised eigenvectors…
I'm trying to find the eigenvector/eigenvalues of the $2\times2$ matrix:
\begin{pmatrix}4 & 2 \\ 2 & 3 \end{pmatrix}
This is my work:
$$\det(A-\lambda I) = \lambda^2-7 \lambda+8=0 \iff ...
1
vote
2answers
50 views
Product of two matrices equals zero
If the product of two $n \times n$ matrices $A$ and $B$ is zero ie: $AB = 0$
Then either $\det(A)$ or $\det(B)$ must be zero.
What additional conditions on $A$ and $B$ would be sufficient ? Clearly ...
1
vote
3answers
62 views
prove or disprove invertible matrix with given equations [duplicate]
Given a non-scalar matrix $A$ in size $n\times n$ over $\mathbb{R}$ that maintains the following equation
$$A^2 + 2A = 3I$$
given matrix $B$ in size $n\times n$ too $$B = A^2 + A- 6I$$
Is $B$ an ...
1
vote
1answer
20 views
Reducing a Laplace Matrix which is also Scalar?
I had a 9x9 matrix as follows: Aim is to reduce it to Lower or Upper Matrix
...
4
votes
6answers
84 views
$(\mathbf{u}^T\mathbf{v})\mathbf{v} = \mathbf{u}^T(\mathbf{v}\mathbf{v})$ doesn't hold for $\mathbf{u}, \mathbf{v}\in\mathbb{R}^n$ - why?
Suppose I have vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^n$. It is well defined to write $5\mathbf{v}$ or $c\mathbf{v}$ for scalar $c$. Since the inner product of $\mathbf{u}$ and ...
1
vote
3answers
51 views
How is my textbook finding this rotation?
I have this transformation $\mathbf x\mapsto A\mathbf x $ which is the composition of a rotation and a scaling. I need to give the angle $\varphi$ of the rotation and give the scale factor $r$. Here ...
2
votes
2answers
47 views
Matrix with orthogonal columns?
What do we call a matrix whose columns are orthogonal, such as $\begin{bmatrix}3 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 1 & 0\end{bmatrix}$?
Is there a special name for it? I tried ...
2
votes
2answers
43 views
Characteristic and minimal polynomial of a special matrix
$H = \begin{bmatrix}
1 & w^{-1} & w^{-2} & ... & w^{1-n}\\
w & 1 & w^{-1} & ... & w^{2-n} \\
w^{2} & w^1 & 1 & ... & w^{3-n} \\
... & ... & ...
0
votes
0answers
18 views
Matrix Algebra in Statistics
We have $y = X\alpha + \xi$, where $\xi\sim N(0, 1)$ and $\xi = (\xi_1\,,\xi_2\,,\xi_3\,,\xi_4)^T$. Moreover $X$ is a 4x4 design matrix such that $x_i$ is a column 4x1 column vector made up of 1's, ...
1
vote
1answer
45 views
How to calculate the determinant using Laplace
How to calculate the determinant using Laplace?
$$
det \begin{bmatrix}
a1 & a2 & a3 & a4 & a5 \\[0.3em]
b1 & b2 & b3 & b4 & b5 \\[0.3em]
...
0
votes
0answers
19 views
3-d Matrices (as in, $A=[a_{ijk}]_{ijk}$)
In programming I've come across the idea of arrays containing arrays containing arrays etc., and as it's pretty intuitive to think of an array of arrays as a matrix, it seems like a reasonable idea to ...
1
vote
0answers
39 views
The meaning of the entries of eigenvectors of graphs
I would like an explanation to the meaning of the different entries of the eigenvectors of a graph.
Furthermore, I'd be happy if anyone can spare an explanation for the meaning of the entries of any ...
3
votes
2answers
62 views
determinant divisible by 13
Question:
Given: $195,403$ and $247$ are divisible by 13.
Prove (without actually calculating the determinant) that
$$\det \begin{bmatrix} 1 & 9 & 5 \\ 2 & 4 & 7 \\ 4 & 0 & ...
2
votes
2answers
61 views
Does the pseudoinverse of $A\in\mathbb{R}^{n\times n}$ commute with $A$ when $A$ is normal?
Suppose that $A^+\in \mathbb{R}^{n\times n}$ is the (Moore-Penrose) pseudoinverse of $A$ and that $AA^t=A^tA$. Does there hold $$AA^+=A^+A?$$
10
votes
5answers
308 views
A matrix satisfying $AB-BA=B$
If $A$ and $B$ are two matrices of $\mathcal{M}_n(\mathbb{R}$) such that
$$AB-BA=B$$
how can we prove that $B$ isn't invertible?
my attempt: I found that $\mathrm{tr}(B)=0$ but I know that this is ...
0
votes
3answers
52 views
Characteristic value or eigenvalues and determinant
I am having semester in linear algebra. And have recently got acquainted to eigenvalues.
What is the relation between eigenvalues and determinant? Going through answers of some questions I found ...
0
votes
0answers
20 views
Prove that if $A$ is symmetric and has a LU-decomposition then $A=LDU' \Rightarrow U'=L^T$, where $L^T$
Suppose the matriz $A$ has a LU-decomposition, in other words, suppose there exists matrices $L$ and $U$ such that $A=LU$ where $L$ is lower triangular and $U$ is upper triangular.
We can to prove ...
1
vote
1answer
62 views
Inverse of matrix sum, special case: $(A + x I)$
Is there a simple way to do $(A + x I)^{-1}$ where $A$ is an invertible matrix, $I$ is unit matrix and $x$ is a scalar?
I see a lot of expressions for the general case $(A + B)^{-1}$, but nothing on ...
-1
votes
3answers
40 views
Questions about matrices and determinants - constant variable multiplication
Is this matrix
$$
M = \begin{bmatrix}
a & -a & a \\[0.3em]
-a & -a & -a \\[0.3em]
a & a & a
\end{bmatrix}
$$
the same as:
$$
M = ...
2
votes
2answers
93 views
How find this matrix has eigenvalues $\lambda_{j}=4\sin^2{\dfrac{j\pi}{2(n+1)}}$
Show that the $n\times n$ tridiagonal matrix
$$A=\begin{bmatrix}
2&-1&0&0&0\\
-1&2&-1&0&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
...
0
votes
0answers
26 views
Are there any properties of the diag operator?
Let $u$ and $v$ be a column vector of same dimension.
1.) Can anyone give some properties about the operations of function, such as $\text{diag}(u)+\text{diag}(v)=\text{diag}(u+v)$?
2.) Is there ...
0
votes
1answer
23 views
Positive semidefinite Matrix examples query
This might be really dumb question but I've just started dealing with such matrices. I would like to know why $$\begin{bmatrix} 0 & 1 \\ 1 & x \end{bmatrix}$$ cannot be a positive semidefinite ...
1
vote
1answer
33 views
Relationship between decompositions of matrices with overlapping kernels
Suppose $A$ and $B$ are two positive semidefinite matrices of size $N\times N$, satisfying
$$
\ker(A) \subseteq ker(B) \quad\Rightarrow \ \ \textrm{Im}(B) \subseteq \textrm{Im}(A)
$$
Further assume ...
1
vote
1answer
47 views
Nilpotent matrices with same minimal polynomial and nullity
From Hoffman and Kunze.
Let $N_1$ and $N_2$ be 6 X 6 nilpotent matrices over the field F. Suppose that
$N_1$ and $N_2$ have the same minimal polynomial and the same nullity. Prove that
$N_1$ ...
2
votes
1answer
32 views
Ask a question about an example in a course note on optimization problem with equality constraint
I have two difficulties on understanding the solution to an example in a course I took this semester on optimization. This example is given to illustrate the usage of Lagrange multiplier method ...
1
vote
1answer
17 views
How to show quantisation error for frequency coefficient
I have the following question from an exam which reads:
For the second question my answer is:
24 5 2 3
9 2 1 3
2 3 2 2
1 1 1 0
I am unsure of what formula ...
2
votes
1answer
31 views
Problem with finding the intersection point between a line and triangle
I have a mathematical problem that I'm trying to solve, but the equations I have derived don't give the correct output when utilised on concrete problems. However, I can't figure out what the problem ...
1
vote
1answer
20 views
Combine transformation matrices
Question:
Find the transformation matrix that combines the following transformation matrices, in order:
$$\begin{bmatrix}
&3 &0 &0 &0 \\
&0 &-1 &0 &0 \\
...
3
votes
3answers
68 views
How to construct a matrix $A$
Construct a matrix $A$ such that $A^2\ne 0$ but $A^3=0$.
I need your help to find $A$. Please help. Thanks in advance.
0
votes
0answers
23 views
Matrix Calculus - differentiate $A\cdot ( x - x^t )$
Having trouble with a derivation - what is the partial with respect to x?
$$ A \cdot( x - x^T)$$
I want to think that it is $A^T-A$, but that doesn't make sense because a non-square A cannot be ...
