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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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Matrix Reduction with Positions

For a matrix $M$ and a set of positions $P=\{(x_i,y_i) \ | \ 1 \leq i \leq n\}$, define $f(M,P)$ as the sum of elements of $M$ at $P$, i.e., $f(M,P)=\sum_{(x_i,y_i)\in P}M_{x_i,y_i}$. Now given $M$ ...
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Matrix perturbation and eigenvector

$Ae=\lambda e$ $e^Te=1$ $A$ is a real matrix. $\lambda, e$ are real. $(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$ Neglecting small terms, $\Delta Ae +A \Delta e=\Delta \lambda ...
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1answer
34 views

How many unique possibilities for $n\times n$ matrix are there?

When given a $n \times n$ matrix (for simplicity let's say $n=3$) how many unique possibilities are there to fill the entries with a set of $m$ numbers (again for simplicity let's say $m=10$). With ...
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1answer
26 views

Does the following result hold?

Suppose that $A$ and $B$ are symmetric and non-negative matrices. Let $\rho(A)$ denote the spectral radius of $A$ and $\rho(B)$ denote the spectral radius of $B$. Does the following result hold? $...
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2answers
27 views

Given path of a unit vector can we derive the rotation matrix?

Let's say a unit vector $\overrightarrow{A}(t)$ in N dimensions is continuously rotated around the origin. It may for example trace out a circle in 3 dimensions or some sort of spiral in higher ...
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Prove that $\langle\mathbf{A}, [\mathbf{C}; \mathbf{0}]\rangle \leq \delta$ equals with $\|\mathbf{A_r}\|_*\leq\delta$ [on hold]

Given an arbitrary matrix $\mathbf{A}\in R^{n\times n}$ and the basis matrix set $\mathbb{S}=\{\mathbf{C}\in R^{r\times n}: \mathbf{C}\mathbf{C}^T=\mathbf{I}\}$. $[\mathbf{C}; \mathbf{0}]\in R^{n\...
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1answer
13 views

Finding matrix from plane in kernal

Give an example of a matrix A such that $\ker(A)$ is the plane $2x − y + 3z = 0$. I am not sure where to start, as I know that the $\ker(A)$ is the matrix of the plane, but I don't know how to go ...
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2answers
27 views

Given two unit vectors what can we say about the rotations that can transform one into another?

Given 2 unit vectors in 2 dimensions there is a uniqe rotation matrix that transforms on to the other. $A$ and $B$ are the two unit vectors is there a general way to find the set of orthogonal ...
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How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?

How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$ here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
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1answer
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Why are singular values of “complex” matrices always real and non-negative?

I've already read the following related questions on math.SE: Why can't singular values be complex numbers? Clarification on the SVD of a complex matrix Why are singular values always non-negative? ...
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What can be said about the definiteness of the following inequality?

Given a Hurwitz matrix $R\in\Re^n$ which has all the eigenvalues located in the closed left-half plane. For a positive-definite matrix $Q$, we know that there exists a unique solution $P$ to the ...
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1answer
33 views

Normalizer in matrix groups

I have the problem of calculating the normalizer of $\begin{bmatrix}      \lambda & 0 \\ 0 & \lambda^{- 1} \end{bmatrix} $ in the group $\begin{bmatrix}      \cos (\theta) & -\sin (\theta) ...
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States of the world/Game theory and Beliefs

This post consists on 3 parts: the question itself, hint and a table. The question will make sense to you only after you have read the tables and the hint attached. The problem is about beliefs of a ...
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2answers
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What is meant by tridiagonal linear equation system?

I have to implement the SOR (Successive Over-Relaxation) method, using sparse matrices, to find the solution vector of these linear equations systems (for quite huge matrices): What does that tridiag(...
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20 views

Matrices up to an equivalence relation

Consider the $n\times n$ matrices over a field $k$, $M_{n\times m}k$. Say that $A\sim B$ for $A,B\in M_{n\times n}k$ if there is an invertible matrix $C\in M_{m\times m}k$ such that $A=BC$. Under ...
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32 views

Matrix of integer powers

Is there a name for the square matrix ($j=0...n$, $k=0...n$) $M_{jk} = j^k$ (with special case $M_{00}:=1$) and is there a closed general formula for its inverse? I have stumbled upon this while ...
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1answer
9 views

Interpretation of vectors in dual forms - in matrix equation, and in linear combination of vectors

While a matrix equation $A \vec x=\vec b$ identifies $\vec x$ and $\vec b$ as two vectors, its equivalent form as linear combinations of vectors ${x_1} \vec {a_1} + {x_2} \vec {a_2} = \vec b$ reveals ...
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2answers
87 views

How to prove that a $3\times 3$ matrix has only $2$ eigenvectors?

I am working through a problem in Riley, Hobson and Bence (Mathematical Methods for Physics and Engineering) that revolves around the following matrix: $$ A= \begin{pmatrix} 2 & 0 & 0 ...
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12 views

Perform Singular Value Decomposition of the following matrix?

Perform Singular Value Decomposition of the following[2*2] matrix and represent in UDV(power)T? [3,2,2; 2,3,-2]
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2answers
57 views

Prove that the equation $ax = b$, $xa = b$, always has a unique solution. [on hold]

A vector space $A$ is called an algebra if in it, in addition to the addition of vectors and multiplication by a number, the multiplication of vectors with properties is defined. In other words, for $...
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22 views

Least squares solution to overdetermined AX=B where each matrix is a rotation

I can't seem to find anything that would help me with this particular problem. I have a bunch of measurements of a matrix A and corresponding matrix B, which I know are related by a third rotation X (...
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1answer
12 views

Most elegant way to show Inner-Product of tensors with indices?

How would you write this sum most elegantly? (Or in other words: What is the most elegant and best way to show a sum of products with indicies?) $\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$ ...
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1answer
36 views

I would like to know how to calculate Matrix

$\begin{pmatrix} x & y+2 \\ 5z & 9k \end{pmatrix}=\begin{pmatrix} 2 & 8 \\ 3 & 9 \end{pmatrix}$ This is calculation of Matrix. $x= \frac{1}{3}, y=y+\frac{1}{4}, z= \frac{5}{3z}, k=...
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1answer
46 views

Decomposing a symmetric matrix as a sum of nilpotent matrices

Assume that a real-valued symmetric matrix $M$ with trace zero can be written as $$ M = A + A^T, $$ with $A^2=0$. Given that $M$ is known, how (if possible) can $A$ be found? The diagonal elements ...
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1answer
46 views

Matrix - Linear algebra problem

Show that there is no matrix $X\in \operatorname{M}_{2\times2}(\mathbb{C})$ with $\operatorname{Tr}(X)=0$ and $e^X=A$, where $$ A=\begin{pmatrix} -1 & 1 \\ 0 & -1 \\ \...
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1answer
18 views

how to apply qr decomposition while keep top left block un-change?

I have a matrix which is already up-triangular like : $\begin{bmatrix}A & B\\ 0 & C\end{bmatrix}$ in which A and C is up-triangular block, now I add a block row then get : $\begin{bmatrix}A ...
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0answers
6 views

Way of predicting change in gradient of line following matrix transformation?

I recently learnt about dominant and repulsive eigenvectors. I noticed that the farther a line is initially from the dominant (though still closer than to the repulsive eigenvector), the more dramatic ...
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1answer
124 views

Matrix representation of Fractional Linear Transformation and the Identity Matrix?

For $x \in \mathbb{R}$, define the fractional linear transformation of $x$ as $f(x)$ where: $$f(x) = \frac{ax + b}{cx+d}$$ Then $f(x)$ has a matrix representation in $\mathbb{R}^2$ of $F$ where: ...
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Are Hypermatrices equivalent to Tensor?

One number $a$ can be seen as a one-dimensional matrix. Can we generalize matrices in a high-dimension sense? Think of a “cubic matrix”, which looks like a crystal with a number attached to its ...
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1answer
36 views

Notation for reversed rows and/or columns of a matrix?

In this answer I am using a transformation of matrices, and I would like to know if there is a notation for this. Given a matrix $A$, let $B$ be the same as $A$ with rows in reverse order, $C$ with ...
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2answers
49 views

$A$ is a square complex matrix and $f\in\mathbb C[t]$ such that $f(A)$ is diagonalizable. If $f'(A)$ is invertible, then $A$ is diagonalizble.

Suppose that $A$ is a square complex matrix and $f$ is a polynomial in $\mathbb C[t]$ such that $f(A)$ is diagonalizable. If $f'(A)$ is invertible, where $f'$ is the derivative of $f$, prove that $A$ ...
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Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
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1answer
21 views

Symmetric Part of Product of 2 tank 2 tensors

Let $A_{ab}$ be a rank 2 tensor. Let $P_{abcd} = A_{ab}A_{cd}$. What are the symmetric and antisymmetric parts of $P_{abcd}$? Is it even possible to define symmetric and antisymmetric parts? (All ...
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25 views

Inequality involving determinant and matrices?

Here is the statement : Let $A\in \mathcal{S}_n^{++}(\mathbb{R})$ and $B \in \mathcal{S}_n^{+}(\mathbb{R})$ then we have the following inequality : $(\det(A+B))^{\frac{1}{n}}\ge (\det(A))^{\...
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0answers
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LU decomposition of matrix product

Let $A_1=L_1U_1$ and $A_2=L_2U_2$ be two matrices with their respective LU-factorizations ($L_i$ is lower triangular and $U_i$ upper triangular). Is it possible to obtain the LU decomposition of the ...
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2answers
38 views

State true or false ( I am not sure what i did wrong)

For 𝐮,𝐯 ∈ ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮+𝐯‖. The dot product of two vectors is a vector. For 𝐮,𝐯∈ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮‖+‖𝐯‖. A homogeneous system of linear equations with more equations than ...
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21 views

If two operators with all eigenvalues real numbers commute then both are triangularizable!

Let $A,B:E\to E$, where $E$ is a finite dimensional vector space, be two linear operators such that all of the eigenvalues are real numbers. If $AB=BA$, prove that there exists a basis in which ...
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1answer
20 views

Equivalence relation on matrices

Consider $M_{n\times n}$ the $n\times n$ matrices over some field $F$. Define an equivalence relation $A\sim B$ if there is invertible $C$ such that $A=CB$. What are the equivalence classes of $\sim$ ...
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1answer
21 views

Parity check matrix H given; justify that C can correct one error, does a word belong to C and correct an error. [on hold]

I'm trying to learn about error correcting codes, and I have this question from a previous exam. Now I'm a little bit confused because I can't seem to wrap my head around this one, could I please get ...
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1answer
15 views

Continuously rotating a unit vector to $e_1$

The following question is from Brian C. Hall's Lie Groups, Lie Algebras, and Representations. Show that $\mathrm{SO}(n) $ is connected, using the following outline. For the case $n =1$, ...
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2answers
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If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$?

I'm a student and I've just read the Characteristic polynomial on Wiki. I have a feeling that: If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$ thanks to the Matrix calculator I've ...
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0answers
54 views

Eigenvalues decrease with power

Take $n \in \mathbb{N}$, and consider a square matrix $A$ of size $n \times n$, with real and positive entries, and such that $\|A\|_2 \leq 1$. I think the following statement holds from simulation, ...
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2answers
71 views

Can a matrix $A$ commute with $e^B$ without commuting with $B$?

As in the title. Is it possible that $[A,B]\neq0$, but $[A,e^B]=0$? I tried expanding the exponential and using $[A,B^n]=\sum_k {n\choose k} B^{n-k}[A,B]B^k $ but this doesn't seem to give any insight....
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3answers
27 views

Prove for complex matrices $A, B$ and $C$, if $AB = BC$ than for each natural number $k$, $\\\\A^kB = BC^k $

in this question and its prove, it is stated that: For complex matrices $A, B$ and $C$, if $AB = BC$ than for each natural number $k$, $\\\\A^kB = BC^k$ I can't see why. Any help would be ...
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Sales analysis for matrix

If the matrix size is 3, 4 and 3 shows product types and 4 shows colour for example. If I need to calculate each products for all the colour, can I? because the row rize is 3 but the colum size is 4 ...
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30 views

Tangent space of matrix group

In Andrew Baker's book 'Matrix Groups', he as defined tangent space to $G$, a matrix group, at $U \in G$ as $$T_U G =\{\gamma ' (0) \in M_n(\mathbb K) : \gamma \ \text{is differentiable curve in} \ G ...
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1answer
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Proving a specific $min$ function is equivalent to solving $Ax-b$

The homework question asks to prove that $min_{x\in\mathbb{R}} {f(x) = 1/2<Ax,x>-<b,x>}$ is equivalent to solving a linear system $Ax-b$. The hint the professor gave is to recite the ...
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27 views

Finding all points in a 2D plane that are k manhattan distance away.

Given an $N * M$ matrix representing a 2D plane and a start point $(sx, sy)$ and another constant $k$ find all points in the matrix such that it has a Manhattan distance equal to $k$. This is how I ...
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0answers
26 views

How to solve a matrix dominated by zeros?

I am trying to solve a matrix of this form: Is there a known algorithm or a method to solve this kind of matrices more efficiently than a normal Gauß elimination method? I input the diagonals as ...
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1answer
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An $n \times n$ matrix $A$ is called skew-symmetric if $A^T = −A$. What values of a, b, c, and d now make the following matrix skew-symmetric? [on hold]

Let $$ A=\left( \begin{matrix} d & 8a-c & 8a+2b \\ a & 0 & 8-5d \\ a+5b & c & 0 \\ \end{matrix} \right) $$ Let $$ A^T=\left( \begin{matrix} d & ...