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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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Symmetry operator matrix over line and plane (reflecting over line/plane) in 3d

Find operator matrix of symmetry over line x = 2y = z and plane containing vectors a = (1, 0, -1), b = (1, 1, -2)
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11 views

How can I find the Schur decomposition from QR factorization?

Let's say that I got a matrix $A$ and I find the $Q$ and $R$ matrix from QR-factorization. $$A=QR$$ Now I want to find the Schur decomposition. $$A=QUQ^{-1}$$ How can I do that? Is it the same $Q$...
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1answer
18 views

How to find the general linear transformation that preserves a certain scalar product?

So I have got the scalar product of two vectors defined in this way: $$(x,y)_c = x_1y_1+cx_2y_2$$ where $x = (x1,x2)$ and $y=(y1,y2)$. Now I need to find the set of tranformation matrices that ...
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0answers
13 views

Specify a unitary matrix for a function

I have a negation function (for bits) that maps a 0 to a 1 and a 1 to a 0. If you translate this into a matrix, it's like this: \begin{pmatrix}0&1\\1&0\end{pmatrix} Now this function is ...
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0answers
15 views

Problem with eigenvalue as a limit

Can anyone point out how to prove this? Let $A$ be a semipositive square matrix, and let $x_m^T [λ_mI-A]=0^T$, where $λ_m$ is the greatest real positive eigenvalue of $A$. Let there be a solution $y^...
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1answer
18 views

Alternative form for U(2) matrix

Do you know how to demonstrate the alternative form that there is in “elementary constructions”? https://en.m.wikipedia.org/wiki/Unitary_matrix
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1answer
33 views

Advice on showing this matrix equality

Let $A$ be an $n \times m$ matrix where the first collumn of $A$ is a $1$ vector So that $A_{i,1}=1 $for $i = 1,2,\dots,n$. Define $G$ as a generalised inverse of the matrix $A^T A$, so that $$A^T A ...
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2answers
45 views

A special subset of $GL(n;\Bbb R)$

Let $M(n,\Bbb R)$ denote set of all $n\times n$ real matrices. For $A\in M(n,\Bbb R)$ we denote $A^t$ as the transpose of $A$. Denote $GL(n,\Bbb R)$ as set of all invertible real $n\times n$ matrices. ...
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0answers
16 views

Extract data from Matlab cell [closed]

As result of calculation I got p as 1x5cell. When I open p every cell contain 93x4 double It means beta=5, so 5 times: 93x4 double, 93x4 double, 93x4 double, 93x4 double, 93x4 double Every of ...
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0answers
22 views

Definiteness of matrix after Woodbury inversion.

Consider a real, symmetric and positive definite $n\times n$ matrix $\mathbf{K}$, and a $n\times m$ matrix $\mathbf{W}$. $\mathbf{W}$ contains $m$ columns with all zeros except a single entry in each ...
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2answers
37 views

Inverse of identity minus matrix exponential

I am trying to analytically find the inverse of a matrix given by: \begin{align} W = \left( I - \alpha e^A \right)^{-1}, \end{align} where $I$ is the identity matrix of appropriate size, $e^A$ ...
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3answers
45 views

$\DeclareMathOperator{\rank}{rank}\rank(A) = \rank(B)$, Prove there exist $U, V$ invertible matrices such that: $A = UBV$

$\DeclareMathOperator{\rank}{rank}$$\DeclareMathOperator{\Mat}{Mat}$Given two matrices $A, B \in \Mat_{m \times n}$ , as $\rank(A) = \rank(B)$. Prove there exist two invertible matrices: $$U \in \...
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1answer
21 views

Is this matrix positive semidefinite (Symmetric matrix, with particular pattern)

Let's consider a symmetric matrix A. If for each row, the diagonal entry is equal or larger than the magnitude of any other element, that is $$a_{ii} \geq |a_{ij}| \quad\text{for all rows } i \text{...
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3answers
54 views

Tensor equation [closed]

If you have two antisymmetric tensors $A_{\mu \nu}$ and $B_{\mu \nu}$, and for every anti symmetric tensor $\epsilon^{\mu \nu}$, $\epsilon^{\mu \nu} A_{\mu \nu} = \epsilon^{\mu \nu} B_{\mu \nu}$ Is ...
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0answers
28 views

Help with this matrix problem about the inverse of a particular matrix

Is the following matrix invertible? $ \begin{bmatrix} x & a & a & \dots & a \\ a & x & a & \dots & a \\ a & a & x & \dots & a \\ \vdots & \vdots &...
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0answers
16 views

Relaxation of a Bilinear Matrix inequality.

I have a Bilinear Matrix Inequality as $$\sum_h F_hx_h+\sum_h \sum_k F_{hk}x_hx_k\leq 0$$ where, $F_h$ and $F_{hk}$ are matrices and $x$ is a vector of appropriate dimension. Now, I have attempted a ...
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1answer
29 views

Smith normal form of the following matrix

Let $$ A = \begin{bmatrix} 66 & 30\\ 12 & 4 \end{bmatrix}$$ I've been trying to find the smith normal form of this matrix, and I keep getting the wrong answer. Here are my workings; gcd of ...
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0answers
52 views

Are Linear Maps resistant to Noise?

Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^*$ which is obtained from the first one plus some ...
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1answer
46 views

Matrix derivatives, problem with dimensions

I'm trying to find a derivative of function: $$L = f \cdot y; f = X \cdot W + b$$ Matrices shapes: $X.shape=(1, m), W.shape=(m,10), b.shape=(1, 10), y.shape=(10, 1)$ I'm looking for $\frac{\partial ...
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0answers
23 views

Matrix [ Eigenvalues and Eigenvectors] [closed]

Greeting I have a question on this Problem What is X2? Thanks you,
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1answer
29 views

$A,B\in \mathbb R^{n\times n}$ share $n$ common linearly-independent eigenvectors $\Rightarrow AB=BA$.

I've been trying to prove the following statement: Let $A,B\in \mathbb R^{n\times n}$ be square matrices such that they share $n$ common linearly-independent eigenvectors. Then $AB=BA$. Everything ...
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1answer
36 views

Regarding a real $n \times n$ matrix $A$ satisfying $A^6 = -A^2$

The Problem: Suppose $A$ represents a real $n \times n$ matrix satisfying $A^6 = -A^2$. (a) Prove that if $A$ is symmetric, then $A = 0$. (b) Prove that if $n$ is odd, then $A$ is not invertible. (...
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0answers
38 views

Intersection of parameterized cosets in a free-abelian group

Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set \begin{equation} S_{\{1,2\}} := \{...
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1answer
41 views

How to properly represent a matrix function.

Given the function $f_{h}(x,y,z)=(x-z,y+hz,x+y+3z)$, what is the correct way to represent the matrix function in respect to the standard basis? With the representation theorem, I would write the ...
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0answers
19 views

how we can make a zero-diagonal Matrix with Unitary matrix

we have a diagonal trace less matrix {like Pauli matrices or gelman matrix $\lande8$ and $\landa3$ in Su(2)} , now we want to apply a unitary matrix on them and make a zero-diagonal and non zero-off ...
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2answers
35 views

Basic linear algebra doubt on Dimension of Vector space

Let $V(R)=M$ of order $2$ $$W_1=\left\{ \begin{pmatrix} a & b \\ 0 & 0 \\ \end{pmatrix} : a ,b \in R\right\}$$ $$W_2=\left\{ \begin{pmatrix} a & 0 \\ c &...
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0answers
33 views

Perform Singular Value Decomposition of (2 0 1;0 1 0)

I've got the questions to perform a singular value decomposition for the matrix A = (2 0 1;0 1 0). I know the method and calculated it to be, A = PDQ, where P = (5 0;0 1) Q = (1/sqrt(5)).(2 0 -1;...
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1answer
30 views

Confusion about the tensor product and matrix multiplication

This is a question I came across looking at special relativity and tensor products. For example, we have the metric tensor and its corresponding matrix representation $$ g_{\mu\nu} = g^{\mu\nu} =\...
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0answers
30 views

Change of eigenvalues of a matrix when pre- and post multiplied by a diagonal matrix

Let $A\in \mathbb{R}^{n\times n}$. Moreover, assume $D$ is an $n \times n$ diagonal matrix with positive diagonals. What is the relation between the eigenvalues of $A$ and eigenvalues of $B:=DAD$? In ...
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0answers
15 views

finding a possible equation systen given a solution set

if given these vectors: vectors ( u,v,w) how can I find an equation systen with this general solution set : solution group
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1answer
42 views

Analytic functions and diagonalisation of matrices.

If I have an analytic function $f$ of a square matrix A (like sin(A)), then I know that if the matrix diagnosable then it is possible to find a matrix $$D = P^{-1}AP \tag{1}$$. Then for a function $f(...
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0answers
17 views

what is the relationship between zeroes in a kernel and the rank of a matrix [closed]

For square matrices, if a kernel has only zeroes, $n$ zeroes, then it is a trivial solution and the matrix has full rank, rank is $n$. But if the kernel has $n-k$ zeroes, does that mean thant the rank ...
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1answer
63 views

Prove that matrix $A$ is symmetric

If $A$, $B$ are square matrices, $B$ is symmetric and $(A+B)^2$ is symmetric, prove that $A$ is also symmetric.
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What is significance of eigenvector of laplacian matrix?

The eigenvector of the Laplacian matrix is widely used in partition technique as spectral graph theory. What is the significance of 'K' eigenvector of the Laplacian matrix?
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Understanding equality regrading expectation of random matrices

I'm reading the following article on Latent Tree Structures (I added a link at the end of the post) : "Spectral Methods for Learning Multivariate Latent Tree Structure". I'm trying to understand the ...
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1answer
18 views

How to find the coordinate vector respect to a matrix basis?

I know how to find the coordinate vector respect to a vector matrix. However, in my textbook, I see the following problem, which asks to find the coordinate vector of M respect to a matrix basis: $M=\...
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0answers
20 views

Inequality with a QR decomposition [duplicate]

Let $A \in \mathbb{R^{n \times n}}$ be an invertible matrix. How to prove $|$det$(A)$|$\leq \prod_{j=1}^{n}{(\sum_{i=1}^{n}{|A_{ij}|^2})^{\frac{1}{2}}}$ with a QR decomposition? I tried to use: ...
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3answers
93 views

How can I construct a nilpotent matrix with the property $A^2 \not= 0$ but $A^3=0$

An example of a matrix $A$ that has the property $A^2=0$ would be $$A= \begin{pmatrix} 0 &1 \\ 0&0\end{pmatrix}$$ However, I can't seem to figure out a "formula" to construct a matrix that ...
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0answers
14 views

Matrix Commutation, is it valid here?

I'm trying to obtain the Ridge Regression solution from the mean of predictive distribution of a Gaussian Process with a linear kernel. The mean of the predictive distribution of a GP is $$\...
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0answers
30 views

Eigen Values for a standard form of tridiagonal Matrices

In my attempt to find a formal solution to a problem in Chemical physics, I have come across a matrix of this form: $\begin{bmatrix} 0 & V_{12} & 0 & \cdots & \...
2
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1answer
45 views
+50

On an inequality involving operator norm of matrices and singular value

Let $A, E \in M_n(\mathbb C)$ be as in this question On invertibility of $A+E$ where $||E||_2<$ smallest singular value of $A$ and $||A^{-1}E||_2<1$ . How to prove that $\dfrac {||A^{-1}b-(A+E)...
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1answer
43 views

Existence of $\lim_{n \rightarrow \infty}A^n$

I was studying Markov chain and in it, it is useful to have a transition matrix $M$ that $\lim_{n \rightarrow \infty}M^n$ exist. So I thought about the existence of $\lim_{n \rightarrow \infty}A^n$ ...
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2answers
38 views

On the entries of $LL^t$ where $L \in GL_n (\mathbb R)$ is lower triangular with positive diagonal entries

Let $L \in GL_n (\mathbb R)$ be a lower triangular matrix with positive diaginal entries and let $A :=LL^t$ . (note that $A$ is positive definite i.e. $A$ is symmetric and all eigenvalues of $A$ are ...
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1answer
24 views

On invertibility of $A+E$ where $||E||_2<$ smallest singular value of $A$ and $||A^{-1}E||_2<1$

Let $A,E \in M_n(\mathbb C)$ . Suppose $\sigma_\min >0$ be the smallest singular value of $A$ and $||E||_2 < \sigma_\min$. Suppose $||A^{-1}E||_2 <1$. Then how to show that $A+E$ is ...
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1answer
17 views

On the sharpness of an inequality involving operator norm of invertible matrices

Let $A\in GL_n(\mathbb R)$. How to show that for every $\delta >0$, $\exists 0\ne r,b \in \mathbb R^n$ such that $\delta=||r||_2/||b||_2$ and $\dfrac {||r||_2}{||A^{-1}||_2||A||_2||b||_2}=\...
2
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1answer
23 views

Partial derivative of coordinates with respect to function

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$. Then $$\frac{\partial f^i}{\partial x^j} = (\nabla f)^i_j$$ where $\nabla f$ is the Jacobian matrix of $f$. When reading this paper I came across the ...
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1answer
40 views

Differentiation of a vector

I have a matrix $A$ which is $n$ by $m$ and a vector $b$ which is $m$ by 1. I have the following expression: $\frac{\delta}{\delta b} \mathbf{1}^{T}f(Ab)$ Where $\mathbf{1}$ is a vector of ones ...
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vote
1answer
36 views

Matrix Multiplication Application

I have a matrix representing the amount of different resources (columns) I would need to create (rows) different objects. $\begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & ...
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votes
1answer
45 views

What is the best algorithm to find the inverse of matrix $A$

I'm going to build a C-library so I can do linear algebra at embedded systems. It's most for machine learning. https://github.com/DanielMartensson/EmbeddedAlgebra/ Anyway, I need to compute inverse ...