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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2
votes
1answer
33 views

Prettifying Bipartite Graph Matrix

Let me tell you the story of this problem. We have $n$ projects and $m$ workers. Each worker can work on multiple projects and each project can be solved by multiple workers. This relationship can be ...
1
vote
3answers
36 views

Find out if there exist such basis where matrix has form

Let $f: \mathbb{C^3}\to \mathbb{C^3}$ has matrix $A=\begin{bmatrix} -1&-2&-3\\0&2&3\\0&-3&-4\end{bmatrix} $ determine if there exist basis where matrix of $f$ has form: $\begin{...
3
votes
2answers
41 views

Torsion of $\mathbb Z ^{2 \times 2}$

I am trying to find the torsion of the ring $R= \mathbb Z ^{2 \times 2}$ seen as an $R-$ left module. So I want to see for which matrices $B$, there exists $A \neq 0$ such that $AB=0$. So if $B=\begin{...
1
vote
1answer
159 views

Obtain all combinations of 3 numbers with repetition.

I'm stuck with this problem and I'd like to get some help. I think there is something I'm not aware of. So, the thing is I'm given this control matrix H. ...
0
votes
1answer
31 views

$2\times2$ normal matrix rotation

I have proved that a $2\times2$ real normal matrix $A$ is either symmetric or of the form $$ \begin{pmatrix}a&b\\-b&a\end{pmatrix} $$ Now, I want to prove that there exists $r>0,\ \theta\in(...
0
votes
2answers
33 views

Determining a $2\times 2$ matrix via its characteristic equation

If I know the following: $A$ is a $2\times 2$ matrix. $A$ has characteristic equation $(\lambda - 2)^2=0$ We can determine that we are dealing with a matrix of the form: $$\begin{bmatrix}2&a\\...
1
vote
2answers
111 views

Can someone provide an example of a collection of matrix that is an open set?

I am totally confused what it means for a matrix to form an open set. Open set to me is either an interval ( , ) in $\mathbb{R}$ or some dotted circle in $\mathbb{C}$ (ok, the dotted circle being $\{z ...
1
vote
1answer
131 views

Computing the matrix that represents orthogonal projection,

There is a theorem that says if $U$ is an orthogonal matrix, i.e., its columns (or rows) form an orthonormal basis, then the action of $UU^T$ represents orthogonal projection of the vector space onto ...
7
votes
2answers
280 views

$3\times 3$ matrix always has determinant $0$. Must $7$ of the elements be $0$?

Let $M$ be a $3\times 3$ real matrix with at least $3$ distinct elements and the property that any permutation of it's elements gives a matrix with determinant $0$. Must $M$ contain exactly ...
0
votes
0answers
45 views

If $P$ is a projection matrix and $PA = PB$, what is the relation between $A$ and $B$?

The projection of the subspace of A and B is the same. This does not necessarily mean A = B.
0
votes
0answers
36 views

Given dynamic system $\dot x = Ax + Bu$, how can we prove that $M$ = {$(A,B)$: system is controllable} is an open set on an Euclidean space?

I wish to show that $M$ = {$(A,B)$: system is controllable} is an open set in some Euclidean space. Equivalently, how can we show that the complement to this set is closed? Here Controllability ...
4
votes
4answers
825 views

Is there a formula for the inverse of Hadamard product?

Say $A$ and $B$ are two square, positive-semidefinite matrices. Is there an expression in terms of matrix product, transpose, and inverse for the Hadamard product $A∘B$? For example, "$(A∘B)^{-1} = ...
1
vote
0answers
53 views

Do tensor norms exist?

Does there exist norms for tensors, as an extension for the ordinary matrix norm? For example, if there is a derivative of a matrix [A] with respect to a vector {x}, does the norm of this derivative ...
0
votes
0answers
21 views

Decomposition of a matrix into the square of another symmetric matrix

I have a got a matrix $M$, is it always possible to find a SYMMETRIC matrix $S$ such that its square equals $M$, namely $S S = M$ and if the answer is positive, what is the procedure to calculate ...
1
vote
0answers
109 views

rank of product of 2 matrices

Please have me with the following question: $A$ is a $m \times n$ matrix with rank $m$, $B$ is a $n \times p$ matrix with rank $p$. Given that $p<m<n$. Is there condition of $A$,$B$ that $\...
1
vote
1answer
73 views

Rank of matrices product with one invertible matrix

Given $A \in\mathbb{R^{7\times8}}$, $B \in\mathbb{R^{8\times8}}$, $C \in\mathbb{R^{8\times7}}$ and provided that B is invertible, how can one check whether $$rank(AC) = rank(ABC)$$ is true? $B$ is ...
0
votes
1answer
29 views

How to check the determinant and rank of multiplied matrices?

Given $A \in\mathbb{R^{7\times8}}$, $B \in\mathbb{R^{8\times5}}$ and $C \in\mathbb{R^{5\times7}}$ How can one check whether $$det(ABC) = 0$$ is true? Given their spaces, the multiplications are "...
0
votes
2answers
38 views

How to perform calculations with unknown matrices

I want to check if there are matrices B,C $\in \mathbb{R}^{3\times3}$ such that $$B^2C^2 - C^2B^2 = A.$$ $A$ is another $3\times3$ matix and looks something like: $$ \left( \begin{array}{ccc} 10 &...
1
vote
0answers
22 views

What is the “minimal” way to turn a Hermitian matrix with zeros on the diagonal into a positive semidefinite matrix?

Suppose one has an Hermitian $n\times n$ matrix, $H$, where every diagonal element is equal to zero. We can add a diagonal matrix $D$ to $H$ such that the sum is positive semidefinite, $H + D \ge0$. ...
1
vote
1answer
32 views

Compute the limit of this expression of norms:

Compute the limit, as n goes to infinity, of the quotient: $$\frac{||A^{n+2}(x)||}{||A^n(x)||} $$, given the matrix $$ \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \...
0
votes
1answer
84 views

solution for Matrix equation

$$ (w*(R_1*P*R_1^{-1})^{-1}+w*(R_2*P*R_2^{-1})^{-1})^{-1}=P_{th} $$ $R_i$ is a rotation matrix 2*2: $$ R_i=\left[\begin{matrix} cos\theta_i & sin\theta_i \\ -sin\theta_i & cos\...
5
votes
0answers
65 views

Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. $\mathcal{M}_{n}$...
1
vote
1answer
82 views

Find if there is a matrix that her sum of each column and row representing two vectors

We have two vectors: $(a_1,...,a_n),(b_1,...,b_m)$. We want to know if there is a matrix $M_{nm}$ that all its elements are from $\left\{0,1\right\}$ with this condition: The sum of all the elements ...
2
votes
0answers
12 views

Find $W\in\mathbb{R}^{T \times N}$ such that for $X\in\mathbb{R}^{T \times N}$: $X'X=W'W-\frac{1}{T}W'\iota_T\iota_T'W$

Given $X\in\mathbb{R}^{T \times N}$, I would like to know how to find $W\in\mathbb{R}^{T \times N}$ such that $X'X=W'W-\frac{1}{T}W'\iota_T\iota_T'W$. $\iota_T$ denotes the vector of ones of length T. ...
5
votes
0answers
2k views

Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) dx dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} 1&4&...
1
vote
1answer
164 views

Relation between number of unique values in Gramian Matrix (G) and the matrices that created it

Lets say I have a $N \times N $ Hadamard matrix $A$ (I am referring to the Kronecker product construction of the Hadamard) and I would like to take the first $M$ rows out of it and then compute the ...
0
votes
0answers
44 views

Product of orthogonal projection operators onto range R$(X)$ and R(X')

Let $X\in\mathbb{R}^{T \times N}$. I am interested in investigating the term $X(X'X)^{-1}X'=(XX^+)(X'^+X')$ where $(\cdot)^+$ denotes the Moore-Penrose-Pseudoinverse. According to Wikipedia these two ...
0
votes
1answer
28 views

The most efficient way to find a minimal value in a 2 dimensional matrix.

Suppose we have a matrix A with value of elements ranging from 0 to 255 ( a greyscale image). What is the most efficient algorithm that will return a minimal value found in this matrix?
1
vote
1answer
23 views

Simple Matrices Help

My lecturer has obviously missed something crucial because this isn't the only question I've been having trouble with. The question: Show that {$u_1,u_2,u_3$} is linearly independent, where: $u_1= \...
1
vote
3answers
66 views

Showing that a bilinear form is non degenerate

Given a finite dimensional vector space $V$ over $F$ and a fixed matrix $(\alpha_{ij})=A \in M_n(F)$ and the bilinear form on $V \times V$ by $B(u,v)=\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha_{ij} \zeta_i \...
1
vote
1answer
55 views

Low-rank matrix space [duplicate]

Let $M(m,n,r)$ be the matrix space of real matrices $m\times n$ with $rank \leq r$. Is $M(m,n,r)$ an open set? or closed set? or Does it have some property? Regards
1
vote
1answer
55 views

Taking limits of norms of a matrix raised to the nth power:

Given a matrix $$ A = \begin{bmatrix} 0 & 3 \\ -2 & 5 \\ \end{bmatrix} $$ and a vector $x = \begin{bmatrix}1&0\end{bmatrix}$, compute $\lim_{n\to\...
0
votes
1answer
33 views

Using the definition of unitary / orthogonal operators explicity for matrices:

If A is unitary, then $$AA^* = A^*A = I, and\ A^* = A^{-1}$$ I want to see this explicitly for a very simple unitary matrix, say, take the column vector A = (1,0,0) and we regard this as a 3x1 ...
0
votes
1answer
41 views

matrix norm equality

I've come across the following equality in a Linear Algebra book: For nonsingular $A \in \mathbb{C}^{n\times n} $, and $w \in \mathbb{C}^n $ $$\max_{w} \frac{\|w\|}{\|Aw\|}=\max_{w} \frac{\|A^{-1}w\...
2
votes
1answer
176 views

How to decompose matrix transformations

Let us assume $A$,$B$ and $C$ are known affine transformation matrices in homogeneous 2D space. If it should happen that $C=A^m B^n$ for some unknown $m,n$, is there a way to detect this short of ...
0
votes
1answer
46 views

How to transform angles to a transformation matrix?

I'm working on an open source project. I need to transform three angles (X, Y, Z) to a matrix. The matrix is a standard 4x3 homogeneous transformation matrix, where the right column describes the ...
1
vote
0answers
78 views

Is there any relation between the Gershgorin circles of a matrix and its resolvent?

Let $A$ be a real symmetric matrix. Now fix a diagonal index say "i" and let $x > max-eigenvalue(A)$. Now is there any thing known about the Gershgorin circle of $[1/(x-A)]_{ii}$ in terms of the $A$...
1
vote
1answer
110 views

Matrix with non-negative eigenvalues

Let $A \in \mathbb{R}^{n \times n}$ be a positive semi-definite $A \succcurlyeq 0$, and with positive diagonal elements ($A_{i,i} > 0$ for all $i$). Let $A$ have at least one eigenvalue equal to $0$...
1
vote
0answers
86 views

Expectation of Projection Matrix

I'm interested in the following question: Let $\mathbf{A}\in\mathbb{R}^{N\times M}$, $M\leq N$ be an random matrix with i.i.d. entries that take strictly positive values. A continuous probability ...
0
votes
2answers
32 views

Checking if a matrix is positive matrix after some coordinate transformations

I have the next positive definite matrix $Q\in\mathbb{R}^{3\times 3}$. For a full rank $U$ (e.g. defining a linear coordinate transformation) I can decompose $Q$ as $$ Q = U^{-1}TU, $$ where obviously ...
0
votes
0answers
42 views

Perspective view and calculating based on it

I've got a project in which i'm asked to use an image captured in perspective view of a lane road (Assume the distance of the camera and the angle relative to the road are known). What I need to do ...
1
vote
1answer
40 views

Set of a matrix

I am working on a homework problem which asks me about the Set of a singular $n\times n$ matrix. specifically whether it is a vector space. I looked in the glossary of the book and searched online and ...
0
votes
0answers
18 views

how can matrix multiplication be described in traffic cordinations?

In one of our lectures, spacial Interaction was represented as matrices. but when It's come to multiplications how would you merge spacial interactions with the result of multiplication. To make ...
0
votes
1answer
42 views

Sum of Matrix Norms and boundedness

Given a matrix $N$, show that there is a constant $C$ such that $$||I + tN + \cdots + t^n N^n|| \le Ct^n$$ for all sufficiently large $t$. I am not sure how to show this. I am guessing I am supposed ...
0
votes
2answers
60 views

Signing of a binary matrix to a totally unimodular matrix

I have the following binary matrix: \begin{pmatrix} 1& 1& 1& 0 \\ 0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ \end{pmatrix} Definition: Signing a matrix ...
0
votes
0answers
31 views

Name for tall zero-one matrix with unit row sum?

Is there a special name for tall zero-one matrix with unit row sum? Call the matrix C. For a data matrix D, the product DC combines columns of D by purely summing without scaling. The rationale is ...
3
votes
1answer
155 views

Characteristic polynomial of a tridiagonal matrix

Consider the polynomial recurrence $$p_{k+1} (x) = (x - \alpha_{k+1})p_k(x) - \beta^2_{k+1}p_{k-1}(x), \quad (k=0,1,\ldots)$$ where $p_0 = 0$, $p_{-1}=0$, and $\alpha_k$ and $\beta_k$ are scalars. ...
0
votes
2answers
447 views

Prove that if A is singular, then adj(A) is also singular

Prove that if A is singular, then adj(A) is also singular. How do you prove this without proving by contradiction?
0
votes
1answer
55 views

Solve this integral of matrix exponential

Let $A \in \mathbb{R}^{n \times n}$ be invertible and define $e^A = \sum \limits_{k=0}^\infty \frac{A^k}{k!}$. Suppose $\lim \limits_{t\to\infty} e^{-A^T A t} = 0$. Show $$A^{\dagger} = \int \limits_{...
-2
votes
1answer
161 views

Calculate nth power of matrix $A$ [closed]

I have a matrix $A$, and need to calculate its $n$th power for a problem. $$A = \begin{bmatrix} a & b \\ d & c \end{bmatrix}$$ Please help me.