Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Geometric interpretations of matrix inverses

Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through ...
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Jacobi's Rotation has two possibilities, why do they both result in same upper triangular magnitude norm?

The Jacobi's rotation is the complex Givens rotation (unitary similarity) that results in a zero for a specified element of a matrix. If the element is not adjacent to the diagonal, then there are ...
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309 views

Trace Minimization of Covariance Matrix

Given a matrix X whose rows contain observations collected at some locations. Can someone explain how trace minimization of covariance matrix $XX^T$ can lead to orthogonal / mutually independent ...
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140 views

how to use determinant calculate the area

fiend the area of the pentagon of the five vertices $(1,2),(4,1),(5,3),(3,7),(2,6)$, please use the way of using determinant. My idea is to cut the pentagon into some triangles, then calculate each ...
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177 views

What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
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finding the largest $p$ components of $x$

Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations: ...
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240 views

What is a good metric to compare matrices?

I have a matrix that I obtained from theoretical computation and I have another matrix which I obtained by actual data collection. How do I compare the two matrices? How do I state that one matrix is ...
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156 views

linear algebra approach to discrete cosine transform

I understand that the discrete Fourier transform simply changes basis to the discrete Fourier basis, which is an orthonormal basis of eigenvectors for any shift-invariant linear operator on $\mathbb ...
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113 views

Curvature of particular Riemannian metric

Let $U = \{ (x_1, \dots, x_n) \mid x_j > 0 \text{ for all } j\}$ and let $\|x\|^2 = \sum_j x_j^2$. The function $x \mapsto -\log \|x\|^2$ is strictly convex on $U$ and thus defines a Riemannian ...
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100 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
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696 views

Good introductory book for matrix calculus

Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
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495 views

What conditions must the constants satisfy so that each of these systems has a solution?

I'm attempting to teach myself linear algebra using this book http://joshua.smcvt.edu/linearalgebra/book.pdf One of the exercises is: ...
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Singular Value Decomposition for Continuous Variables

say I have a $n\times n$ matrix $w_{ij}$. I can preform a singular value decomposition such that $w_{ij}=\sum_l \sum_n u_{il}\lambda_{ln}v_{nj}$ with $\lambda_{ln}$ diagonal. Now, is there such a ...
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211 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
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173 views

Problems on Symmetric Matrices

1 . Let $A = (a_{ij})$ be a real $n \times n$ matrix such that $a_{ij} = a_{ji}$ for all $1 \leq i,j \leq n$ and $a_{ij} = 0$ for $|i-j|>1$. Moreover $a_{ij}$ is non-zero for all $i$,$j$ satisfying ...
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358 views

Nullspace of matrix with multivariate polynomial entries

Let $R:=\mathbb{Z}[X_1,X_2,\dots,X_{mn}]$. Suppose $A=(f_{ij})$ is a $m\times n$ matrix with entries in $R$ such that (1)there is no zero column in $A$; (2)for each $i,j$, either $f_{ij}=0$ or ...
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Generating rotations in $\mathbb{R}^n$

I want to be able to computationally generate a rotation matrix for $\mathbb{R}^n$ where $n$ might go as high as $10^4$. The naive technique would be to generate the rotation in each plane then ...
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171 views

Solving a system with present pre-multiplication

Suppose a symmetric matrix $L\in\mathbb{R}^{n\times n}$ is given, and a rectangular matrix $A\in\mathbb{R}^{n\times m}$, $m<n$. A solution to the system $$LAx=b, \tag 1$$ is sought, for known ...
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175 views

Matrix riddle I came across

I came across this matrix riddle a couple of weeks ago and I haven't figured it out. You have a known $10\times10$ matrix $A$, which is symmetric. For some unknown transformation matrix $T$, you ...
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144 views

Positive definite completion of a matrix

Suppose we have a real, symmetric matrix $A(x_1,x_2,x_3)$ given by \begin{pmatrix} a_{1,1} & a_{1,2} & x_1 & x_2 \\ a_{2,1} & a_{2,2} & a_{2,3} & x_3 \\ x_1 & a_{3,2} & ...
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A question of permanent function

Let $A, B, A-B$ be positive definite matrices. How to show $\mathrm{per} A\ge \mathrm{per} B$? Here $\mathrm{per}$ is the permanent function. Also, if $A$ is $n\times n$ doubly stochastic matrix ...
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Nested “Formal” Block Determinants

I found this interesting formula on Wikipedia on blocked determinant of a square matrix: $$\begin{vmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{vmatrix}= \begin{vmatrix} ...
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Computation of determinant of a matrix with elements from an arbitrary commutative ring

The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I ...
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266 views

Finite dimensional function space “different” from $\mathbb{R}^n$ generically

If you pick a random vector in $\mathbb{R}^n$ with some fixed basis, there is no special relationship between components. The relationship between the $1^{st}$ component and the $5^{th}$ component is ...
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196 views

“Shrinking Map” Problem for Normed Space

Today I thought of a question. Which normed vector spaces (over $\mathbb{R}$) have the following property? For all $\epsilon >0,$ there exist nonzero $\delta \in V$ and continuous $f_{\delta }$ ...
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1k views

Light Out Puzzle Solution

I am decided to solve the puzzle game named Lights out. So, i choose linear-algebra to solve my problem, so note that this link, i start my work as follow : NOTE : Any light states can accept two ...
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99 views

Enforcing sum of Hermitian matrices to be positive definite

I have a set of Hermitian matrices $A_i$ which are orthonormal in the Frobenius inner product $\langle A_i,A_j\rangle=\mathrm{Tr}(A_i^{\dagger}A_j)=\delta_{ij}$, and I know there exists a basis in ...
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182 views

Does matrix convergence in $L^p$ imply convergence of the eigenvalues in $L^p$?

Let $A_n(x)$ be a sequence of symmetric matrix functions that converges in $L^p(\Omega)$ to $A(x)$. Is it true that the eigenvalues of $A_n(x)$, or a subsequence of these, converge to the eigenvalues ...
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48 views

Maping non-linear constraint to linear subspace

What I will ask, more than a solution, is the correct definition of my problem and directions to find the solution. I have a set of linear equations, e.g.: \begin{eqnarray*} d_1 =& L_1 - 9\,m_1 ...
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A solution to a system of homogeneous polynomials in three variables

If one has (say) five degree $3$ homogeneous polynomials $f_1,f_2,f_3,f_4,f_5$ in three variables $x,y,z$, and $f_j(x_0,y_0,z_0)=0$ for all $j$ for some fixed $(x_0,y_0,z_0)$, can we conclude that ...
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156 views

How to invert this function on matrices which involves the permanent?

I'm interested in understanding whether a particular natural function on matrices, closely related to the permanent of a matrix, is invertible, and whether its inverse admits a simple closed form. The ...
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Research Problem on Abstract Algebra popping up from Control Theory

Here is a problem that came up in my research: Let $X,U$ be finite dimensional vector spaces over some field $F$ and let $f: X \rightarrow X, \beta : U \rightarrow X, \kappa : X \rightarrow U$ be ...
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357 views

Why is Cholesky factorization numerically stable

It's often stated (eg: in Numerical Recipes in C) that Cholesky factorization is numerically stable even without column pivoting, unlike LU decomposition, which usually need pivoting schemes. But ...
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563 views

General properties of eigenvalues of a Jacobian matrix when premultiplied by a symmetric, positive definite matrix?

For a particular engineering problem that I'm working on, I have computed a Jacobian matrix $J$ and there is another matrix $M$ associated with the problem. $M$ is known to be symmetric, real-valued, ...
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335 views

If $S$ and $T$ are commuting, normal operators, then $ST$ is normal

If $S$ and $T$ are commuting, normal operators, then $ST$ is normal That says it all, but let me be more specific. (By the way Wikipedia says this: "The product of normal operators that commute ...
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Rank of matrices

A is an $m \times n$ matrix. If $\operatorname{rank}(A)=n$ and $AX=0$ where $X$ is $n \times k$, then $X = 0$. Below is how I conclude $X = 0$. Since $\mathsf{rank}(A)=n$ and $A$ is $m \times n$, so ...
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245 views

How to generate an $n \times n$ rotation matrix?

It is well known that the $2 \times 2$ rotation matrix is given by, $$\left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array} \right]$$ and ...
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121 views

Sparse matrix inverse multiplied by sparse matrices

I have the equation $\bf E = Y D^{-1} Y^\top$. $\bf D$ is a potentially large sparse $m \times m$ matrix, and $\bf Y$ is a sparse $n \times m$ matrix, where $n \ll m$. Is there a particularly ...
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254 views

basis free volume form for a symplectic vector space

It's easy to show, using a symplectic basis, that if $\omega$ is a symplectic form on a $2n$-dimensional vector space $V$, then $\omega^n \neq 0$. I'd like to be able to prove it without choosing a ...
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Question about ring and module

Consider a tangent bundle with even and odd parts $T_0 + T_1$, define a space $\Omega^{k,l}_{p,q}$ consisting of (p,q)-forms taking values in $\wedge^k T_0 \otimes \wedge^l T_1$, i.e. the space of ...
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193 views

About Invariant Factors

Suppose $A$ is a $2\times2$ matrix with minimal polynomial $x^2- 5x+4$, $B$ is a $2\times2$ matrix with minimal polynomial $x^2 -6x +8$, and let $O$ be the $2\times2$ matrix with all entries $0$. ...
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$XX^t=A$, $X=?$. Where $X \in \{0,1\}^{n \times m}$

The problem: $XX^t=A$, $\quad$ ($X_{ij}\in{0,1}$, $\quad$ $\sum_{j=1}^m x_{ij}=2$), $\quad$ $X=?$ Details: $n,m \in N$ $A \in \{0,1,2\}^{n \times n}$ $X \in \{0,1\}^{n \times m}$ $A$ is a ...
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Uniqueness of the unbounded solution for linear equation

Let $\mathbb X$ be the set of all vectors $x$ such that $x_i\in(-\infty,\infty]$ for all $1\leq i \leq n$. Let $A,b$ be a matrix and a vector with non-negative real entries (bounded) and consider the ...
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Taking Square Roots of Matrices over Z/nZ

Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n? If the matrix is diagonalizable, then does diagonalizing and taking the square roots work? ...
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143 views

On the distribution of unimodular matrices generated by the Hermite normal form

A problem I'm currently considering requires me to generate (pseudo-)random Gaussian integer matrices with Gaussian integer matrix inverses. By analogy with an algorithm I know for generating random ...
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128 views

Linear algebra: Alternated multilinear forms

Let $A^{\#}:A_{n}(F)\rightarrow A_{n}(E)$ define by $A^{\#}f(v_{1},\cdots ,v_{n})=f(Av_{1},\cdots, Av_{n})$, where $A_{n}(F)$ is the space of the alternated n multilinear forms in F. Verify that ...
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142 views

Generalization of Plücker embedding

Let $V$ be a vector space and $1 \leq k \leq n$ natural numbers. By $\operatorname{Grass}_n(V)$ I mean the Grassmannian of $n$-co​dimensional subspaces of $V$, that is, $n$-dimensional ...
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212 views

Basic question on Linear algebra

Let $V$ be an inner product space and $v_1,\ldots,v_n\in V$ be basis with $(v_i,v_j)\leq 0$ for $i\neq j$. Suppose that there exists vectors $v_1^*,\ldots,v_n^*\in V$ satisfying ...
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109 views

Proving that the image of the tangent map is a subspace

Let $X$ and $Y$ be Euclidean spaces and consider the tangent map of a smooth map $f:X\rightarrow Y$, $T_p f: T_pX \rightarrow T_{f(p)}Y$, defined by $(p, v) \mapsto (f(p), \partial f(p)v)$. The author ...
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Cramer's rule for infinite dimensional vectors

For the equation $Ax = b$ in the finite dimensional linear space one can apply Cramer's rule to find $x$ if operator $A$ is linear. If there is an equivalent or a similar method for an infinite ...