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), ...

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1
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0answers
14 views

Gradient calculation for Matrix

I have the following: $$b^T a^{-1} b$$ what is the gradient wrt to $a$. $a$ is matrix and $b$ is vector. Basically I should take the derivative with respect to $a$. Is it correct that it equals to: ...
-3
votes
0answers
13 views

MATLAB -Banded LU Factorization No pivoting [on hold]

Please show MATLAB code and explain all steps. I know that I need to make q and p arguements, but I am not sure how to make sure there's no pivoting and the bandwidths exist. I'm not sure how to ...
1
vote
0answers
24 views

Let A be a 2 by 2 matrix with real entries

A = $$ \begin{matrix} a & b \\ c & d \\ \end{matrix} $$ a.) Prove that we have $ A^2-(a+d)A + (ad-bc)I_2 = O_2$ b.) Show that if there exists an integer ...
0
votes
0answers
10 views

How to simulate the stroboscopic effect? [on hold]

Can anyone show me an example of estrobocópico effect in matlab? Or at least send me a link of any simulation. Thank you.
4
votes
2answers
39 views

Necessary and sufficient conditions for when spectral radius equals the largest singular value.

One well known fact about matrix norms is the following: If $\lambda_1\geq \dots\geq \lambda_n$ are eigenvalues of a square matrix $A$, then: $$\frac{1}{||A^{-1}||} \leq |\lambda|\leq ||A||$$ If we ...
0
votes
0answers
15 views

Find Homogeneous System from Solution Spaces

I have the following vectors in the Subspace of $\mathbb{R}^5$ $U=\mbox{Span}[(1,-1,-1,-2,0), (1,-2,-2,0,-3), (1,-1,-2,-2,1)]$ $W=\mbox{Span}[(1,-2,-3,0,-2), (1,-1,-3,2,-4), (1,-1,-2,2,-5)]$ I need ...
0
votes
0answers
15 views

Method for determining next pivoting row?

Given systems of linear equations like $A$, how does one computationally find out what order to place the rows to ensure that no diagonal elements become zero during Gaussian Elimination? (If the ...
1
vote
2answers
37 views

If A is regular, then $AA^T$ is positive definite, since $x^TAA^Tx=(A^Tx)^T(A^Tx)>0$

I read this statement and didn't understand why the right part of the equation is true. Namely, that: $(A^Tx)^T(A^Tx)>0$ Can someone explain? Thank you.
3
votes
1answer
16 views

Density of orthogonal matrices with rational coefficients

Is it true that ${O}_n({\mathbb R}) \cap {\mathbb Q}^n$ is dense in ${\cal O}_n({\mathbb R})$ for any $n\geq 2$ ? It obviously suffices to consider the density of ${SO}_n({\mathbb R}) \cap {\mathbb ...
0
votes
1answer
27 views

Inverse matrices properties.

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ ...
-1
votes
0answers
34 views

Can't find a diagonal dominant matrix

I tried to solve system of linear equations using Jacobi method and one of the step is diagonally dominant matrix. My initial matrix is: $\begin{bmatrix}11.07 & 8.01 & -8.47 & 6.84\\16.65 ...
0
votes
1answer
22 views

Which complex vector multiplied by its conjugate returns the identity matrix

I am trying to find (in case there is any) which complex vector $n$ of 2 dimensions, multiplied by its conjugate transpose, returns a diagonal matrix. $n = [a, b]^T = [a_1+ja_2, b_1+jb_2]^T$ ...
1
vote
1answer
19 views

show inverse by matrix multiplication

Suppose $v^Tu \neq 1$ and $u,v \in \mathbb{R}^n$. Both $u$ and $v$ are column vectors. Define matrix $A=I+uv^T$. Show by matrix multiplication that $$A^{-1}=I-\frac{uv^T}{1-v^Tu}$$ My attempt: ...
0
votes
1answer
19 views

Multiply image kernels

If I have 3 image kernels (edge detection, sharpen, box blur) and want to perform each to an image in that order. Can I multiply the 3 together in some way and apply them to my image in one go? Here ...
-3
votes
1answer
24 views

Coset Leaders and Syndromes

This is my Parity check matrix For Coset Leader 100010 Syndrome is 101 Could any one help me with the procedure, since I figured ...
0
votes
1answer
15 views

Find the Basis and Dimension of a Solution Space for homogeneous systems

I have the following system of equations: x+2y-2z+2s-t=0 x+2y-z+3s-2t=0 2x+4y-7z+s+t=0 Which forms the following matrix ...
0
votes
2answers
29 views

Proving antisymmetry within matrices [on hold]

If $A$ is a $3\times 3$ antisymmetric matrix of real numbers, how can I prove that $A^2$ is a symmetric matrix?
0
votes
1answer
17 views

Effect of the nature of noise on the spectrum of a random matrix

Consider the following two equations $X = M + \eta_1$ $Y = M + \eta_2$ where, $X\in\mathrm{R}^{n\times n}$, ia a real random matrix with mean $M\in\mathrm{R}^{n\times n}$. $\eta_1$ is Gaussian ...
-1
votes
2answers
32 views

Proving a matrix is always symmetric [duplicate]

$B$ is a square matrix of real numbers. Show that the matrix $BB^T$ is always symmetric.
-1
votes
0answers
9 views

World- Camera-Projector coordinates using respective extrinsic matrices

When calibrating a camera using a library such as OpenCV, you get as a result the intrinsic matrix, distortion coefficients and numerous extrinsic matrices depending on the number of "chessboard ...
2
votes
3answers
117 views

Skew Symmetric Matrix Properties

We have a theorem says that "ODD-SIZED SKEW-SYMMETRIC MATRICES ARE SINGULAR" . Proof link is given here if needed. Now let us assume we have a $3\times 3$ skew symmetric matrices of the form $ ...
1
vote
0answers
31 views

Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
0
votes
0answers
23 views

Quaternions- Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represnts the position vector as result of rotation with an angular veclocity $\omega(t)$ in quaternion , then you can make the relationship ...
1
vote
1answer
8 views

Inverse rotation transformations

I'm taking the 2-degree gibmle system and position its alignment point in a arbitrary position (denoted by the axes angles phi for the first degree, and theta for the second). How can I reverse the ...
0
votes
1answer
24 views

Multiplication of Rotation Matrices in quaternion

Given Data and specifications NB : * means multiplication Suppose we need to rotate a point $P = \begin{pmatrix} x\\ y\\ z \end{pmatrix}$ with rotation matrix ${Q}_{3\times3}$ then what we do is ...
-2
votes
0answers
26 views

matrices and eigen values [NBHM-2014] [on hold]

In each of the following cases, describe the smallest subset of $\Bbb{C}$ which contains all the eigenvalues of every member of the set $S$. a. $S = \{A ∈ M_n(\Bbb{C}) | A = BB^*, B \in ...
0
votes
1answer
12 views

Matrix norm square properties.

I'm trying to prove one of these inequalities. This isn't a homework problem but trying to solve out of curiosity as it didn't have any relationship between $x$ and $\alpha$. How do you prove: ...
2
votes
3answers
83 views

Getting back the column vector from which a matrix was generated

We know that given a vector $\mathbf{X}\in \mathrm{R}^{n\times 1}$, we can create a matrix $\mathbf{A} = \mathbf{XX}^T$, where $\mathbf{A}\in\mathrm{R}^{n\times n}$. Now let us suppose the reverse ...
0
votes
0answers
30 views

Eigenvalues and Determinants of Two Matricies

Suppose $B=[v,e]$ is an $n \times 2$ matrix with $v=[v_1,...,v_n]^T$ and $e=[1,...,1]^T$, and $J_{2\times 2}=[(0,1),(1,0)]$, and so $Rank(BJB^T)=2$. How can we prove that $BJB^T$ and $JB^TB$ have the ...
0
votes
2answers
25 views

Show that if matrix $A$ is symmetric, then so is $P^TAP$.

I need to show that if $A$ is symmetric, then so is $P^TAP$, assuming the matrix multiplications are valid. I'm sure if I actually expanded the matrices to show the entries and did the ...
2
votes
2answers
60 views

Determinant of the sum of an identity matrix and a rank-two-symmetric matrix

Suppose $I$ is an $n \times n$ identity matrix, and $S$ is the $n \times n$ symmetric matrix with rank equals two. I was reading something saying that: $$\det(I-S)=(1-\lambda_1)(1-\lambda_2)$$ where ...
1
vote
2answers
31 views

Solving equation different solutions

The vector $b=\pmatrix{x1\\x2}$ gets roated with a matrix about $30 \deg $ what results in the vector $ \overline b =\pmatrix{6\\8} $ Now my task is to find the original vector coordinates of $b$ ...
1
vote
1answer
28 views

Proofing that an matrix is idempotent

My task was to show that certain matrices are idempotent, that is, ${AA} = {A}$. I struggled a with the proof for one case and when I allok at the solution, I have problems understanding onse step. ...
0
votes
0answers
13 views

Application of Frobenius inequality [on hold]

What are the most interesting applications of the Frobenius inequality?
0
votes
1answer
31 views

For what values of $k$ will these equations have no solution/infinite solutions/unique solution

Here are the 3 linear equations: $$x+y-z=-1$$ $$2x-4y-6z=-1$$ $$x-y+(k^2-1)z=k$$ I understand a $4\times3$ matrix must be set up in order to solve this particular problem.The part which I get ...
3
votes
0answers
29 views

Easy elementary proof of Farkas Lemma?

Is there any elementary proof of Farkas lemma which does not use convex analysis and hyperplane separation theorem? What about special case below: If the Matrix $A$ is invertible, then there is ...
0
votes
1answer
25 views

Optimising using Hessian matrix

I am bit perplexed in optimisation problem if the principal minor is zero. If the principal minor is zero does it mean that the Hessian matrix is always indefinite and the point of extremum will refer ...
0
votes
0answers
10 views

How to calculate discrete cosine transform for a matrix

I have a 8x8 matrix and I want to calculate its discrete cosine transform (DCT-II). I have this formula but I don't know to use it with a matrix. In the French Wikipedia they gave an example for ...
1
vote
0answers
13 views

Identities for the Hilbert–Schmidt norm of products of projections.

I've been studying different metrics on the Grassmannian $Gr(k,n)$ of k-dimensional linear subspaces of $\mathbb{R}^n$ and found myself needing some identities for the norm of a product of orthogonal ...
0
votes
0answers
8 views

Open Leontief input output model

An economy is divided into three sectors: Manufacturing, Agriculture and Services. For each unit of output, Manufacturing needs 0.1 units of Manufacturing, 0.3 units of Agriculture, 0.3 units of ...
0
votes
0answers
19 views

Changing a negative definite matrix to a positive definite matrix

Consider a negative definite matrix $X$,then $(I-e^X)$ is a positive definite matrix. What condition should matrix $X$ satisfy?
0
votes
4answers
38 views

commutative matrix multiplication of nxn matrices?

If there are two matrices A and B that are both nxn matrices, will AB = BA always? Is there a way to have those two matrices so that AB = 0 but BA ≠ 0?
0
votes
4answers
45 views

Finding the diagonalizing matrix.

Find a nonsingular matrix $C$ such that $C^{-1}AC$ is a diagonal matrix. $$ A=\begin{pmatrix} 1 & 0 \\ 1 & 3 \\ \end{pmatrix} $$ I have found the eigenvalues to be 1 ...
1
vote
0answers
23 views

Closed-form expression for this matrix equation?

I have the following matrices $P \in \mathbb{R}^{N \times N}$, $q(k) = \begin{bmatrix} q_1(k) \\ \vdots \\ q_N(k) \end{bmatrix}$. With $q_i(k) \in \mathbb{R}^n$ and thus $q(k) \in \mathbb{R}^{Nn}$. ...
17
votes
2answers
917 views

Are $10\times 10$ matrices spanned by powers of a single matrix?

I don't know how to answer this question: Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the ...
3
votes
1answer
49 views

Matrix manipulation using trace

Suppose that $u$ is an $N\times 1$ random vector and $M$ is an $N\times N$ nonrandom positive semi-definite matrix that is also idempotent: $M\times M=M$. Claim: $E(u'Muu'Mu)=\text{Tr}\{M ...
0
votes
1answer
42 views

If A is a matrix, what does A' mean?

If A is a matrix, what does A' mean? I have tried google this but nothing came up. My new stats course had some review problems, and these multiple choice came up. Which statement is true? (a) ...
4
votes
1answer
40 views

Are primitive row stochastic matrices diagonalizable?

Let $A$ be an $n \times n$ matrix with real, non-negative entries. Assume $A$ is primitive, meaning there exists an integer $k$ such that $A^k>0$ (here the inequality means all entries in $A$ are ...
0
votes
1answer
20 views

$A$ and $B$ conjugacy

Show that the matrices $A=\begin{pmatrix}2&0\\0&0\end{pmatrix}$ and $B=\begin{pmatrix}2&0\\1&0\end{pmatrix}$ are not $\mathbb{Z}$ conjugate (there exists no matrix ...
1
vote
1answer
23 views

Trace of vectors

Does that sound about right? Given that x is $m\times 1$ and y is $m\times 1$ vectors, show that $ tr(\mathbf{xy'})=\mathbf{x'y}$. Attempt: By using the property of ...