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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1answer
66 views

Can a elementary row operation change the size of a matrix?

My question is very simple - Can an elementary row operation change the size (eg: $2\times2$ or $3\times 2$) of a matrix? I think the answer should be no, but while reading Linear Algebra by Hoffman ...
0
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0answers
16 views

Control System (block reduction & mason's rule)

i am trying to simplify this block diagram. I calculated something but I am not sure about it, is my reduction correct? Thank you. [![here is my question][2]][2] This is my answer
0
votes
1answer
10 views

Value of $a$ if system of equation is consistent.

If the following equations are consistent and have more than one solution, what is the value of $a$? Given $u+v=-(av+1)$ $u+2v=-a(v-1)$ $3u+8v=a+2$ I was thinking that system of equation is ...
1
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0answers
7 views

Equivariant matrices and commutation relations

Let $T_1,T_2\in R^{d\times k}$ matrices and $G$ a finite unitary group of cardinality $N$. Indicating (a matrix representation of the) elements of $G$ with $g$, equivariant matrices can be written ...
0
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0answers
33 views

Matrices inequality

Let $A$ and $B$ are $m\times n$ and $n \times m$ matrices, respectively and $AA^T\leq I_m$ and $BB^T\leq I_n$ (i.e. $AA^T-I_m $ and $BB^T-I_n$ are negative semi-definite), where $I_m$ and $I_n$ are ...
0
votes
0answers
14 views

Probability matrix from a adjancency matrix

i have this adjacency matrix of 3 nodes: |0 1 0| |0 0 1| |1 1 0| Now i need to find the associated probability matrix. Naturally i would say it would look ...
0
votes
2answers
42 views

Find all similar matrices to diagonal matrix

The given task is to find all 2x2 Matrices A that are similar to: a) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ b) $\begin{bmatrix} 1 & 0 \\ 0 & 1 ...
10
votes
1answer
96 views

Fake proofs using matrices

Having gone through the 16-page-list of questions using the tag (fake-proofs), and going though Best Fake Proofs? (A M.SE April Fools Day collection) and ...
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0answers
28 views

If $AS$ is traceless for every $S$ skew hermitian then $A=0$

In a paper I'm reading, we're given three $n \times n$ matrices, $B,Y,S$ where $S$ is skew hermitian. Given: $tr([B,Y]S) = 0$, the paper concludes that $[B,Y] = 0$, but gives no proof. How do I prove ...
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2answers
26 views

Is there any way to know the algebraic multiplicity of the $0$ eigenvalue in the minimal polynomial when the rank is $1$?

Say I have a matrix $A$ of $r=rank(A)=1$ I know that in the characteristic polynomial the algebraic multiplicity of $(\lambda-0)$ is $n-r$ which in my case is $n-1$ Is there a rule about the ...
1
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1answer
29 views

Matrix reaised to an exponent

$If\quad the\quad matrix\quad A\quad =\quad \begin{bmatrix} 1 & \quad -1 \\ -1 & \quad \quad 1 \end{bmatrix}\\ \qquad Then\quad { A }^{ n+1 }\quad =\quad ?$ My effort So i tries tried ...
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0answers
12 views

Inverse of a rectangular matrix with positive elements

In general a rectangular matrix $(m\times n)$ with all positive elements may have moore-penrose g-inverse whose all elements need not be positve. Is there is any special structure of $(m\times n)$ ...
0
votes
1answer
61 views

Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then ...
1
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1answer
13 views

How do I rearrange an adjacency matrix of an acyclic digraph so its non-zero elements are above the diagonal?

Any graph can be represented by an adjacency matrix. The matrix for an acyclic digraph can be represented as a matrix with all its non-zero elements above the diagonal. However, if I were to take an ...
3
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0answers
22 views

Determinant of a large block matrix

$\newcommand{\lmt}{\left[\begin{matrix}}$ $\newcommand{\rmt}{\end{matrix}\right]}$ Hi, I was reading through a proof of the number of domino tilings of a $(2n)\times(2n)$ chessboard, and somewhere ...
3
votes
1answer
52 views

Elegant proof of an elementary result in Linear Algebra

I've been reading Hoffman Kunze, and I came across this theorem (theorem $9$) which has a long and tedious proof. I've been wondering wether there could be a more elegant proof. Theorem 9. Let $e$ ...
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0answers
8 views

Solving for homography - SVD vs linear least squares (Matlab)

so I had an assignment in Matlab for solving for a homography (and stitching images) and I solved it by converting the coordinates into homogeneous form (since scale doesn't matter in our assignment) ...
0
votes
1answer
19 views

Prove that $\text{det}(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$

Let $f(x)=(p_1-x)\cdots (p_n-x)$ $p_1,...p_n\in \mathbb R$ and let $a,b\in \mathbb R$ such that $a\neq b$ Prove that $\text{det} A={bf(a)-af(b)\over b-a}$ where $A$ is the matrix: ...
1
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0answers
22 views

Notation: rotation matrix with a condition

I'm building a space simulation & am using this resource for converting Keplerian Orbit Elements to Cartesian Co-ordinates. The notation for step 6 has me slightly confused: Is the top part ...
2
votes
1answer
30 views

Prove that $\lambda_1^2$, $\lambda_1\lambda_2$ and $\lambda_2^2$ are eigenvalues of matrix $A$

This is the problem I am currently having trouble with: If $\lambda_1$ and $\lambda_2$ are eigenvalues of matrix $$ \begin{bmatrix} a & b\\ c & d\\ ...
1
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1answer
90 views

Eigenvectors of “weighted” Hermitian matrix?

Consider two real matrices $\boldsymbol{H}$ and $\boldsymbol{D}$ with the following properties: $\boldsymbol{H}$ is a symmetric matrix (since it is a real matrix this is equivalent to being ...
0
votes
2answers
21 views

Moving vectors to the left and the right of a product

Suppose that $A$ and $B$ are $1\times n$ row vectors and $x$ is a $n\times 1$ column vector. I have an expression $$ (Ax)^2B'B $$ which is an $n\times n$ matrix. Question: Is it possible to write ...
0
votes
1answer
67 views

Matrix-rank changes after rotation

I might be wrong. Assume matrix $M$ as a data matrix (e.g., a 2D image). $M$'s rank represents the underlying dimension of the data (or the degree of freedom). For example, $M$ usually isn't ...
0
votes
1answer
11 views

Decomposition of a positive semidefinite matrix

Let $Y \in \mathbb{R}^{n \times n}$ be a symmetric, positive semidefinite matrix such that $Y_{kk} = 1$ for all $k$. This matrix is supposed to be factorized as $Y = V^T V$, where $V \in ...
0
votes
1answer
20 views

square matrices A and B have equal rows/colomns and A*B = I matrix does that mean that B*A also = I? [duplicate]

If you have two square matrices with equal rows and columns A and B and AB = the identity matrix does that mean that BA also equals the identity matrix?
9
votes
1answer
83 views

Prove the n-th power of a matrix is the null matrix

Let $A,B$ squared matrixes with complex elements, $dim(A)=dim(B)=n, AB=BA, \det(B)\ne0$, having the following property: $|\det(A+zB)|=1, \forall z \in \mathbb{C}, |z|=1$. Prove ...
0
votes
3answers
50 views

Express eigenvectors of $A^{-1}$ in terms of eigenvectors of $A$

I know the eigenvalues of the matrix $A^{-1}$ are $\frac{1}{\lambda_n}$ where $\lambda_n$ are the eigenvalues of $A$. I didn't know their eigenvectors were related; in what way are they related? Also ...
2
votes
1answer
59 views

Prove that $R(+,.)$ is a division ring but I disproved it

QUESTION: Let $R=\left[\begin{matrix}\alpha & \beta \\ \bar\beta & \bar\alpha\end{matrix}\right]\in \mathbf{M_2(\mathbb{C})} $ where $\bar\alpha,\bar\beta$ denote the conjugates ...
16
votes
1answer
310 views

What is the number of $n \times n$ binary matrices $A$ such that $\det(A) = \text{perm}(A)$?

Recall that the permanent is the 'positive analog' of the determinant whereby the signs in the cofactor expansion process are taken as positive. That is, the permanent is the immanant corresponding to ...
0
votes
0answers
47 views

Does $B^2 \leq A^2$ imply $\| A^{-1} B\| \leq 1$ for the operator norm?

Assume we have two $n \times n$ real symmetric matrices $ A^2 $ and $B^2$, such that it holds for some $0\leq\rho<1$ $$ 0 < (1-\rho)B^2 \leq A^2 \leq (1+\rho)B^2, $$ where "$\leq$" means ...
1
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0answers
14 views

Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
0
votes
0answers
62 views

Proving that eigenvalues are positive iff $\det(A_k)> 0$ for all $k = 1, …, n$ for a real symmetric matrix $A$

I am trying to prove that eigenvalues of $A$ are positive iff $\det(A_k)> 0$ for all $k = 1, ..., n$ for a real symmetric matrix $A$ where $A_k$ is the $k \times k$ matrix obtained by deleting the ...
1
vote
1answer
33 views

Existence of Unimodular Congruence Transformation for Symmetric, Integer matrices

Two symmetric, integer valued matrices, $K_1$ and $K_2$, are congruent if there exists a unimodular integer matrix, $X$, such that $$X^T K_1 X = K_2$$ What are the conditions on the existence of such ...
0
votes
0answers
18 views

$A^{\pi}$ property

Can someone give an example for the matrix mentioned in the following definition. Definition is taken from "Iterative Solution Methods" of Owe Axelsson. Definition: The matrix $A$ said to have ...
1
vote
1answer
20 views

Algorithm to check if number x exists in matrix

I have the task to develop an algorithm which checks if a specific number x exists in an int-array[][]. Further the 2-dim array entries have following terms: $$array[i][j] \leq array[i][j+1] \space ...
1
vote
2answers
29 views

Eigen-values of a matrix $P^{-1}AP$

QUESTION: If A and P be $2$ non-singular $n\times n$ matrices and $\lambda$ is the eigen-value of $A$, then show that $\lambda$ is also the eigen-values of a matrix $P^{-1}AP$. I could simply ...
0
votes
1answer
20 views

Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and ...
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0answers
14 views
1
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0answers
16 views

Proving that an element of a ring annihilates a module [duplicate]

Let $R$ be a commutative ring with $1$, $M$ be a finitely generated $R$-module, $\mathfrak{i}$ an ideal of $R$, and $\phi$ an $R$-homomorphism such that: $$1.\;\phi(M)\subseteq \mathfrak{i}M=M$$ i. ...
4
votes
1answer
502 views

Assistance with proof of $(AB)^T=B^T A^T$

If $A$ is $m \times n$ and $B$ is $n \times p$ matrices, prove that $(AB)^T = B^T A^T$. Matrices' elements are $A = [a_{ij}], B = [b_{ij}]$. Let $C=AB=[c_{ij}]$, where $c_{ij} = \sum_{k=1}^n ...
1
vote
1answer
30 views

Effect of simple linear transformation

Consider the linear transformation given by $$T\left\{\begin{bmatrix}x \\y\\z\end{bmatrix}\right\} =\begin{bmatrix}-x\\y\\z\end{bmatrix}$$ Find a matrix $A$ such that $T(x) = Ax$, where x = $[x, y, ...
0
votes
0answers
19 views

Rank of a symmetric matrix after removing a column and row.

If I have a $n\times n$ symmetric matrix $M$ with real entries, zeros on the diagonal, and two of the column vectors are identical and I remove one of these columns, and the corresponding row, then ...
1
vote
1answer
43 views

Does $A^2 \geq B^2 > 0$ imply $ACA \geq BCB$ for square positive definite matrices?

Assume we have two $n \times n$ real nondegenerate matrices $ A^2 $ and $B^2$, such that $$ A^2 \geq B^2 > 0, $$ where "$\geq$" means positive semidefinite (Loewner) ordering. Does the following ...
1
vote
1answer
445 views

Computational complexity of Gaussian elimination

If it took me approximately 4 minutes to solve an equatian $Ax=b$ for $x$ (where $A$ is a $3\times3$ matrix and $b$ is a $3\times1$ matrix) using Gaussian elimination, how much longer would it take me ...
6
votes
2answers
172 views
+50

Kernel of a Vandermonde like matrix

I am wondering how to show that the following matrix has trivial kernel: $$\begin{bmatrix} 1&1&1&1&1&1 \\ s_1&s_2&s_3&s_4&s_5&s_6 \\ ...
0
votes
0answers
23 views

Find $B\leq 0$ such that $A(B-I)\geq 0$ for a given $A$ symmetric p.d.

All the inequalities are to be understood component-wise Here are two strongly related questions; in both cases, I am looking for conditions on matrix $B$ which I could deduce from the assumptions. ...
1
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1answer
33 views

Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So ...
3
votes
1answer
22 views

Determinant of hankel matrix of hyperbolic functions, $a_n=\frac{n}{\sinh(\pi n)}$

I am trying to learn about the properties of Hankel matrices, and they appear to have nice closed forms for quite a large class of sequences. The class I am interested is when the elements $a_n$ are ...
0
votes
2answers
35 views

Generate random variables without cholesky decomposition

How would you generate n standard normal random variables with $n\times n$ correlation matrix $R = (r_{ij} )$ where $r_{ij}$ is $1$ if $i = j$ $\rho$ if $i \neq j$, with $\rho \geq 0$ without ...
2
votes
3answers
3k views

Trace of the matrix power

Say I have matrix $A = \begin{bmatrix} a & 0 & -c\\ 0 & b & 0\\ -c & 0 & a \end{bmatrix}$. What is matrix trace tr(A^200) Thanks much!