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
27 views

Factorization of $\left( \begin{array}{cc} A+D&\vec x \\ \vec y^T& a \\ \end{array} \right)$.

Let $A$ be a $(n-1)\times (n-1)$ matrix, $\vec x,\ \vec y$ be two vectors of dimension $=n-1$, $a$ be a real number. Let $D$ a $(n-1)\times (n-1)$ diagonal matrix. From which, matrix $$\left( ...
0
votes
1answer
26 views

Which of following option(s) are correct for symmetric matrix with real entries

Let $A = [[a_{ij}]]$ be an $n \times n$- non-singular symmetric matrix such that each $a_{ij}$ is a positive integer. Then we can conclude that (A) the determinant of $A$ is a positive integer (B) ...
0
votes
0answers
21 views

How to judge the convexity of this function?

$ f(X) = -\log \det(X^TX+I)$, $X \in \mathbb{R}^{n \times n}$, is this function convex or not? Does anybody have an idea about this problem? Thanks.
0
votes
0answers
52 views

How to reach this result using only matrix operations?

$$\mathbf{Xw}-\lambda_w \mathbf{1}=0$$ Where $\lambda_w$ is a scalar, $\mathbf{w}$ is a vector, and $\mathbf{X}$ is a symmetric p.d. matrix. It is known that $\mathbf{1'w}=1$ (but I don't think that ...
6
votes
1answer
72 views

Does anyone know the Burnside Matrices?

For $G$ a fine group with conjugacy classes $C_1,\dots C_k$ we introduced the Burnside Matrices $A_r$ where $1<r<k$ with entries: $$A_r := \Big(\sqrt{\frac{|C_t|}{|C_s|}}a_{rst}\Big)_{1\leq ...
1
vote
1answer
36 views

Converting a matrix from one base base to another.

I have this basis $B = ((1,0,1),(0,1,-1),(1,-1,0))$ That is represented by: $$[T]_B = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 2 \\ 1 & 1 & 2 \end{pmatrix}$$ I want to convert ...
0
votes
1answer
37 views

Prove/disprove: if $X$ is an eigenvector of $T$ then X is a singular matrix

I have this question: Let $A$ be a non-scalar matrix of an order $(N\times N)$. And let $T:M_{n\times n}^R \rightarrow M_{n \times n}^R$ such that: $T(X) = AX$ for every $ X \in M_{n\times n}^R$ ...
3
votes
1answer
51 views

Eigenvalues and eigenspaces of AB

Problem: Consider two matrices $A, B \in \mathbb{R}^{3 \times 3}$. Suppose $A$ has three distinct real eigenvalues $\lambda_1, \lambda_2$ and $\lambda_3$ with respective eigenspaces $E_{\lambda_1}, ...
1
vote
3answers
45 views

Obtaining a Non-Singular Matrix from a Singular one by Perturbation

In a paper "http://www.math.cornell.edu/~nussbaum/papers/08-1.pdf" (page 264 Lemma 2) I encountered the following way of obtaining an invertible (non-singular) matrix from a non-invertible (singular) ...
0
votes
1answer
33 views

$\|AA^Tx\|_2\geq\sigma_1\|A^Tx\|_2$ with $\sigma_1$ being the smallest positive singular value of $A$?

Regarding the $\ell_2$-norm, we know that $$\|A^Tx\|_2\geq\sigma_0\|x\|_2,$$where $A$ is a matrix, $x$ is a column vector, and $\sigma_0$ is the smallest singular value of $A$. Now consider ...
3
votes
0answers
56 views

Contractible pieces of $GL(n,\mathbb{C})$

Is $GL(n,\mathbb{C})$ contractible for any $n$? My intuition is telling me it is not, because the determinant maps the general linear to $\mathbb{C}\setminus 0$ which is not contractible. If there ...
1
vote
1answer
89 views

Powers of a $2 \times 2$ matrix [closed]

Let $A$ be a $2 \times 2$ matrix such that $$A = \begin{pmatrix} \sin\frac{\pi}{18} & -\sin\frac{4\pi}{9} \\ \sin\frac{4\pi}{9} & \sin\frac{\pi}{18} \end{pmatrix}$$ Find the smallest number $n ...
2
votes
2answers
35 views

Property of 10x10 matrix with non negative eigenvalues

Let $A$ be a $10\times 10$ matrix with complex entries such that all its eigenvalues are non-negative real numbers, and at least one eigenvalue is positive. Which of the following statements is ...
2
votes
3answers
153 views

A question on commutation of matrices

Given a diagonal matrix $D$, and a nilpotent matrix $N$, do we always have $DN=ND$? If not so, what further conditions do we need to have it? This question came form an ODE/Linear Algebra problem: ...
0
votes
1answer
24 views

Measure of independency of vectors in a full rank matrix

Suppose A1 and A2 are two full rank matrices of similar size. What could be the parameter which say that one of matrix have more independent vectors compared to another matrix? In other words, column ...
0
votes
0answers
20 views

Property of invertible matrix [duplicate]

I just need a hint to start thinking. Thanks
0
votes
1answer
37 views

Find new coordinates after rotating a shape centrally

i want to find new points of shape in Cartesian coordinate system after applying an angle on shape. What is formula to find these points
0
votes
0answers
13 views

What do we know about rank-2 perturbations?

Are there any theorems known about the changes in spectrum of a matrix A when it is changed to A+X, when X is rank-2? I am particularly interested in the case when X is a zero matrix except for ...
2
votes
0answers
32 views

A special matrix

In matrix $\textbf{A}=[a_{kj}]_{K\times K}$, each elemtent on the main diagonal is $a_{kk}=1$. Other elements is $0\leq a_{kj}\leq1$. Besides, each non-diagonal elements satisfies $a_{kj}\geq ...
2
votes
3answers
35 views

Finding two linearly independent solutions for a homogeneous linear system

I'm having difficulty getting the same answer as a textbook solution to a problem. The basis of the problem is a finding two linearly independent solutions to a homogeneous linear system of the form: ...
0
votes
1answer
87 views

Prove that $T$ is not diagonizable

I'm having difficulties with this exercise, can anyone give me a hand? Let $T:R^3 \rightarrow R^3$ be a linear transformation. It's know that $(1,1,0), (1,1,1)$ are eigenvectors of $T$ and: ...
0
votes
1answer
28 views

What does the notation $M^{[2]}$ mean with regards to matrices?

I am busy studying transitive closures of relations. The Matrix of the relation, $M_R$ is $$M_R = \begin{pmatrix} 1&0&1\\ 0&1&0 \\ 1&1&0 \end{pmatrix}$$ As you might know ...
2
votes
1answer
34 views

Linear algebra: Gram-Schmidt process and QR factorization of a matrix

I first start out with the following matrix: \begin{pmatrix} 1 & -2 & -5 \\ 0 & 1 & 3 \\ -5 & 2 & 2 \end{pmatrix} I then used the Gram-Schmidt process to get: ...
0
votes
0answers
22 views

Is there an efficient $O(n^2)$ way to estimate in MATLAB a matrix condition number given its LDL decomposition?

Since evaluating a matrix condition number usually takes $O(n^3)$, I wonder whether there is an efficient $O(n^2)$ way to estimate in MATLAB a matrix condition number given its LDL decomposition. ...
1
vote
1answer
41 views

Kernel of a specific antisymmetric matrix

I am trying to compute the kernel of the following real antisymmetric $m \times m$-matrix: $A = (a_{ij})$, where $a_{ij} = \begin{cases} 0, & i = j, \\ \lambda_{j-i}, & j > i, \\ ...
0
votes
1answer
19 views

Remove scale transformation from a complex transform matrix 4x4?

My common task is I have a rect with coordinates of its $2$ points: $(x, y, z), (x + a, y + b, z)$. I applied a $4\times4$ transform matrix to it and it became a quadrilateral. Now for some reasons I ...
1
vote
1answer
45 views

Is there a name for the matrix constructed in this way. Does it have any other interested properties?

Recently I participated in a competition, where I was given two vectors $X$ and $Y$. A matrix is constructed from this two vectors in such a way that $$M_{i,j} = \frac{1}{x_i + y_j}$$ The task was ...
2
votes
1answer
31 views

Eigenvectors of a matrix with complex entries

I'm trying to prove that if $\vec{z}$ is an eigenvector of a matrix $A$ with complex entries, then $\bar{\vec{z}}$ is an eigenvector of $\bar{A}$. My approach: $A\vec{z}=\lambda\vec{z} \implies ...
0
votes
2answers
28 views

Rewrite matrix equation as a quadratic programming problem

Given real-matrix $X_{n\times p}$ how can the problem of minimizing $Tr(X^TA_{n\times n}X)$ under the constraint $Tr(X^TC)=\phi$ be posed as a standard convex quadratic program given by the form: ...
1
vote
2answers
67 views

Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$

Given an arbitrary $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$? Update: ...
0
votes
1answer
31 views

Any square matrix can be row operated into Triangular Matrix

I am doing a proof for a quesition, which requires an assumption that any square matrix can be transformed in to Triangular matrix using row operation How do I go about proving such obvious ...
0
votes
0answers
22 views

Elements of SO(n) is block-diagonalizable

I am not able to show that elements of SO(n) are conjugate to a block-diagonal matrix with 2x2 blocs that are rotation matrices, and a 1x1 bloc 1 if n is odd. Can someone help me please?
0
votes
1answer
40 views

Is there a deterministic way to find such a matrix $X=AS$ basic linear algebra

I have a matrices $X \in \mathbb R^{512 \times 768}$, $A\in \mathbb R^{512 \times 50}$. I'm looking for a matrix $S\in \mathbb R^{50 \times 768}$ such that $X=AS$. Is there an algorithm that does ...
1
vote
1answer
38 views

How to prove $(I-e^{At})^{-1}$ only contains positive element?

Given a symmetric $N\times N$ matrix $A$, with eigenvalues $-x_1,-x_2,-x_3,\dots,-x_N$ and $x_1,x_2,\dots,x_N >0$. $A$ is known as a negative-definite matrix. We can diagonalize $A$ as $A = ...
9
votes
4answers
609 views

Is every noninvertible matrix a zero divisor?

Is every noninvertible matrix over a field a zero divisor? Related to this: What are sufficient conditions for a matrix to be a zero divisor over a noncommutative ring?
1
vote
2answers
35 views

Prove that $A$ is diagonalizable and find similar matrices

Let $A$ be a matrix $(3x3)$ such that: $A(1,1,1)^t=(2,2,2)^t$ and $rank(2I + A) \lt rank(2I-A)$ I need to prove that $A$ is an diagonalizable matrix and find all the matrices that are similar to it. ...
0
votes
1answer
43 views

Eigenvalues of Inner Product Matrix

Matrices have a least two major functions in linear algebra. On one hand, they can represent linear transformations as elements of $\text{Hom}(V,V)$). On the other hand they can represent inner ...
1
vote
1answer
42 views

Generalized partial trace

I am interested in finding a general rule (from the matrix point of view) for calculating the partial trace. Starting from a matrix $$ A = X_1 \otimes X_2 \otimes \cdots \otimes X_n $$ I know how to ...
8
votes
3answers
382 views

Is there a name for this type of vector norm?

In the case of the $\mathcal{l}_2$ norm we have, $$||\mathbf{x}||_2^2=\mathbf{x}^T\mathbf{x}.$$ I was wondering if there was a type of norm that had a linear operation embedded in it, like this, ...
0
votes
5answers
34 views

Determining a matrix from its definition.

Suppose that $u$ is a $n \times 1$ matrix such that $u^Tu=1$. Determine the matrix $A=I-2uu^T$. I have taken some specific examples and the answer is always $-I$. Is this true in general?
0
votes
0answers
12 views

3D analogue to following matrix transformation template?

What would be the analogous form for the transformation matrix of a 3-dimensional shape as opposed to the 2-dimensional shape form presented below? My impression is that the analogue would be a 4*4 ...
1
vote
1answer
17 views

Does there exist any non-trivial square matrices of dimension $n$ with power cycles of less than $n$

Earlier I was faced with the matrix: $$A=\begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}$$ Which cycles ...
1
vote
0answers
16 views

Finding point closest to origin on a hyperboloid

(1) Let A be 3x3 real symmetric matrix. The eigenvalues of $A$ are $\lambda_1 = -6, \lambda_2 = 1, \lambda_3=4$ $q(x_1,x_2,x_3) = -x_1^2 + x_2^2 -x^2_3 + 10x_1x_3 = 1$. $A$ is the matrix of $q$. I ...
0
votes
1answer
30 views

Nonconstant solutions of discrete predator and prey model and Perron-Frobenius

Consider the discrete dynamical system given by $x_{n+1} = A x_n$, where $A = \begin {pmatrix} a & -b\\c &d\end {pmatrix}$ and $x_n = \begin {pmatrix} u_n\\v_n\end {pmatrix}$. Are there ...
3
votes
0answers
28 views

Question concerning a correspondence between basis elements of the Schur algebra and some matrices

I have the following question: Let $k$ be an infinite field and let $S_k(n,r):={A_k(n,r)}^{∗}=\text{Hom}_k(A_k(n,r),k)$ and $A:=A_k(n):=\text{polynomial functions on}\ \Gamma:=\text{GL}_n(k)$ and ...
0
votes
1answer
16 views

Is it possible to solve for scalar in this multiplication of two quadratic forms involving inverse matrix?

Given the following two quadratic forms: $$a^2=\mathbf{w'Xw}$$ $$b^2=\mathbf{1'X^{-1}1}$$ And the known relations: $$a^2b^2=1$$ $$\mathbf{X}=\mathbf{\Sigma}-\lambda\mathbf{R}$$ Where ...
0
votes
1answer
40 views

Condition to Guarantee $n$ Distinct Eigenvalues

I looked around, and as far as I can tell, I haven't found this question anywhere else on SE, so if I somehow missed it, please pardon me. I think this is probably a standard result, but I am having ...
0
votes
1answer
194 views

how to prove if $\det A=\det B$ then $A=CB$?

Let $A$ and $B$ be invertible $n \times n$- matrices and $C$ be an $n \times n$- matrix with $\det C =1$. Prove that $\det A = \det B$ if and only if $A=CB$. I've got the proof backward but I got ...
1
vote
4answers
28 views

Associated matrix with respect of a basis

Given the following linear transformation: $$f : \mathbb{R}^2 \to \mathbb{R}^3 | f(1; 0) = (1; 1; 0), f(0; 1)=(0; 1; 1)$$ find the associated matrix of $f$ with respect of the following basis: $R = ...
0
votes
1answer
18 views

$T: \mathbb C ^{n \times n} \to \mathbb C ^{n \times n}$ defined by $T(A)=BA$, so every eigenvalue of $T$ is eigenvalue of $B$

Let $T: \mathbb C ^{n \times n} \to \mathbb C ^{n \times n}$ defined by $T(A)=BA$. $A,B \in \mathbb C ^{n \times n}$. Prove: Every eigenvalue of $T$ is eigenvalue of $B$, and ...