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
3answers
57 views

Determine whether a set of 4, 2x2 matrices form a base for M2.

I am having a hard time solving this question: Let $A,B,C,D,E$ be $2\times2$ matrices above R field. If $\{\,AE,BE,CE,DE\,\}$ linearly independent then $E$ must be an invertible matrix. it feels ...
0
votes
0answers
35 views

Fibonacci sequence matrix

Let us recall that the famous Fibonacci sequence: $0,1,1,2,3,5,8,13,21,\dots$ is defined as follows: we put $\phi_0 = 0, \phi_1 = 1$ and define $\phi_{n+2} = \phi_{n+1} + \phi_n$. We want to find a ...
0
votes
0answers
29 views

How to prove a function is not positive definite [on hold]

I have a lecture about matrix analysis. I have already know some strategies to prove that the function is positive definite. But I face difficulties when I try to see that the (bounded) function is ...
1
vote
0answers
52 views

Perturbation of the principal eigenvector of a PSD matrix

Setting: I have a $n \times n$ PSD matrix $A$ and $\tilde{A}=A+E$ be its symmetric perturbation such that $\|E\|_2=\epsilon.$ Let $(\lambda,u)$ be the principal eigenvalue, eigenvector pair of $A$ and ...
1
vote
1answer
36 views

Is there a simpler matrix for these rotation?

I'm still in high school so sorry if I do not know this. I learnt matrices in class and how to use them to rotate by $90^o,180^o,$ and $270^o$ with center (0,0). I played around with them later and ...
2
votes
3answers
62 views

What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?

Consider a full rank $n\times n$ symmetric matrix $A$ (coming from a set of physical measurements). I do an eigendecomposition of this matrix as $$A = E V E^T$$ Most of the eigenvalues are positive, ...
0
votes
0answers
17 views

ALS Matrix Convergence

I came across this statement in some MapR documentation on ML in Spark: ALS is an iterative algorithm. During each iteration, the algorithm alternatively fixes one factor matrix and solves for ...
4
votes
3answers
137 views

Doubt with vectorial spaces (Basis and dimension)

Good night, i'm working in a problem, i need an basis and the dimension of the space. $a_{1}=(1,0,0,-1),\:a_{2}=(2,1,1,0),\:a_{3}=(1,1,1,1),\:a_{4}=(1,2,3,4),\:a_{5}=(0,1,2,3)$ I make this: $\left[ ...
2
votes
3answers
40 views

Row replacement operation not changing the determinant

Can someone prove why a row replacement operation does not change the determinant of a matrix? **row replacement operation being adding one row to another or something of that sort
0
votes
1answer
22 views

Upper bound of a bilinear form

Suppose I have a form $|X^TBY|$ where $X \in R^n, Y \in R^m$ and $B \in R^{n \times m}$ is a matrix whose elements are bounded. Is there an upper bound for the whole expression of the form $|X^TBY|\le ...
1
vote
3answers
33 views

Matrix and scalar multiplication

Say we have the following variables: A, a matrix that is nxn in size containing complex numbers B, a matrix that is also nxn in size containing complex numbers x, a scalar If you multiply, does it ...
0
votes
1answer
21 views

How to reduce a matrix algebraic expression with a division?

I'm not really sure about the algebra rules here, but it feels like my expression should be reducible. Say I have a 1xN vector called w. Then say I have an NxN matrix called M. Say I have the ...
0
votes
0answers
21 views

Let $A\in M_{n \times n}(\mathbb{C})$, $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a C.P. for $g(A)$.

Let $A\in M_{n \times n}(\mathbb{C})$. Its characteristic polynomial $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a characteristic polynomial for $g(A)$. I believe ...
0
votes
0answers
20 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
16 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
vote
0answers
16 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
votes
0answers
19 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 ...
-1
votes
0answers
38 views

Matrices inequality [on hold]

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
2answers
53 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 ...
1
vote
1answer
33 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 ...
0
votes
0answers
17 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 $(m\times n)$ ...
10
votes
1answer
139 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 ...
1
vote
3answers
50 views

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

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
vote
1answer
68 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
vote
1answer
24 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 ...
5
votes
1answer
44 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
57 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 $e$ ...
0
votes
1answer
24 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
vote
0answers
25 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
33 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
vote
0answers
9 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
2answers
23 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 ...
1
vote
1answer
20 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
21 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?
1
vote
1answer
107 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 ...
4
votes
1answer
80 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
votes
3answers
56 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 ...
1
vote
0answers
24 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
19 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
24 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 ...
2
votes
1answer
65 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 ...
9
votes
1answer
89 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 ...
1
vote
2answers
31 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
25 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 ...
-1
votes
0answers
16 views

Find the center of Group in 2 cross 2 matrices in group theory [on hold]

Find the center of Group in 2. 2 matrices
1
vote
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. ...
1
vote
1answer
35 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
22 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
34 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, ...
3
votes
1answer
26 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 ...