Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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2
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1answer
490 views

is there a generalization of unimodular matrices for non-square matrices?

Is there a generalization of unimodular matrices for non-square matrices? It is well-known that unimodular matrices guarantee an integral solution for a linear program (if the constraint matrix is ...
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3answers
251 views

figuring things out in linear algebra

There are things in linear algebra that I would like to better understand, from an intuitive point of view. For example, matrices are entities that may be use to transform a domain into another, by ...
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1answer
21k views

Properties of a matrix whose row vectors are dependent

When a column vector in a matrix is a made up of "combination" of its other column vectors, it is said to be linearly dependant. Say... $$ A=\begin{bmatrix} 2 & 1 & 0\\ 4 & 5 & -6\...
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5answers
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Why Aren't “Similar” Matrices Actually the Same?

In linear algebra, a matrix $B$ is said to be "similar" to $A$ if $B=C^{-1}AC$, that is $B$ = a matrix $A$ multiplied by a third matrix $C$, and its inverse, $C^{-1}$. In regular algebra, if I take a ...
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1answer
2k views

Understanding and using the transfer-matrix-method

Let $G = (V,E,\Phi)$ be a weighted directed graph and $\mathcal{W}' : E \rightarrow \mathbb{C}$ the weighting. Let additionally $m = \# V$, $E_m$ the $m \times m$ identity matrix. Let $v,w \in V$ be ...
25
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5answers
25k views

Can a real symmetric matrix have complex eigenvectors?

A Hermitian matrix always has real eigenvalues and real or complex orthogonal eigenvectors. A real symmetric matrix is a special case of Hermitian matrices, so it too has orthogonal eigenvectors and ...
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4answers
4k views

Eigenvectors of a normal matrix

According to the spectral theorem every normal matrix can be represented as the product of a unitary matrix $\mathbb{U}$ and a diagonal matrix $\mathbb{D}$ $$\mathbb{A} = \mathbb{U}^H\mathbb{D}\...
4
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2answers
2k views

Derivative of inverse quadratic function of a matrix

I have been stuck with the following derivative for some time: $$ \frac{\partial\,\mathbf{b}^\mathrm{T}(\mathbf{X}\mathbf{C}\mathbf{X}^\mathrm{T})^{-1}\mathbf{b}}{\partial\,\mathbf{X}} $$, where $\...
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5answers
27k views

Linear Algebra: determine whether the sets span the same subspace

So I am stuck on 51 here: 51. Determine whether the sets $S_1$ and $S_2$ span the same subspace of $\mathbb{R}^3$: $$\begin{align*} S_1 &= \Bigl\{ (1,2,-1),\ (0,1,1),\ (2,5,-1)\Bigr\}\\ S_2 ...
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2answers
3k views

Deriving the Derivative of the “Normal” Vector in the Frenet Formula

In the Frenet Formula, I can see why the derivative of the tangent vector is a function of the curvature times the normal vector, and the derivative of the binormal vector is a negative function of ...
6
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3answers
430 views

Linear algebra: rank

Let $A:E\rightarrow F$ be a Linear Transformation between finite dimensional vector spaces, with $\mathrm{Rank}(A)=r$ and $\dim E=n$, $\dim F=m$. Prove that there are basis in $E$ and $F$ such that ...
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4answers
66k views

Getting the inverse of a lower/upper triangular matrix

For a lower triangular matrix, the inverse of itself should be easy to find because that's the idea of the LU decomposition, am I right? For many of the lower or upper triangular matrices, often I ...
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2answers
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Linear Algebra: linear combinations problem

So I have problem number 4: So what I tried doing is that I set the vector from a) equal to c1*(first vector from S) and c2*(second vector from S), and from that I got the 4x3 matrix: [6,4,-42;-7,6,...
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4answers
3k views

Matrix Representation of Inner Product

In my readings I have on several occasions encountered references to a linear algebra theorem that runs as follows: Let $g$ be a non-degenerate inner product on the real vector space $V$. Then, there ...
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2answers
994 views

Example of eigenvectors in different bases (follow-up question)

This is a follow-up question on this one: Connection between eigenvalues and eigenvectors of a matrix in different bases Assume I have matrix $$ B=\left( \begin{array}{cccc} 0 & 0 & 1 & ...
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3answers
20k views

Connection between eigenvalues and eigenvectors of a matrix in different bases

If you have a matrix $A$ you can find its eigenvalues and eigenvectors. If you represent this matrix relative to another basis $\mathcal{D}$ you can again find its eigenvectors and eigenvectors. My ...
6
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1answer
1k views

Involuted vs Idempotent

What is the difference between an "involuted" and an "idempotent" matrix? I believe that they both have to do with inverse, perhaps "self inverse" matrices. Or do they happen to refer to the same ...
3
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1answer
325 views

Linear algebra: Projections

Let $P$, $Q:E\rightarrow E$, be projections and $PQ=QP$, show that $N(P)+N(Q)=N(PQ)$, $N(P)$ stands for Kernel of $P$ As $P$, $Q$ are projections and $PQ=QP$ then $PQ$ is a projection, so $E= N(PQ)\...
3
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4answers
173 views

Orthonormal basis error diagonally dominant?

I'm working on an error estimate for a numerical method, and in the process I've stumbled across the following abstract inequality which I think is true, but am having a hard time proving. Suppose $\{...
5
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1answer
316 views

Proving that $\det{(A^i_j})= \sqrt{ |\det{(G)}|}$

Let $V$ be an $n$-dimensional vector space and let $(v_1, \dots, v_n)$ denote any oriented basis for $V$. Also, let $g$ be an inner product on $V$ and let $G$ denote the Gram matrix of inner products $...
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1answer
1k views

Linear Algebra: Vector Subspaces problem

I am currently stuck on number 16: So theorem 4.5 says: If $W$ is a non empty subset of a vector space $V$, then $W$ is a subspace of $V$ if and only if the following closure conditions hold: If $\...
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1answer
13k views

How do I determine if the vectors lie on a plane in an $N\times N$ matrix?

In a $2\times2$ matrix, it is quite easy to see if the vectors lie on a plane or not. By vector, I mean the columns of the matrix. I usually determine if the numbers are of a certain multiple. From ...
5
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1answer
140 views

A problem in Linear algebra

Suppose $A$ is a $n$ by $n$ matrix with entries $a_{ij}$ such that $$|a_{ii}|>\sum_{k\neq i}|a_{ki}|$$ for $i=1,2,...,n$, prove $A$ is invertible.
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2answers
52 views

Finding $a_n$ using a given matrix

I want to find a formula for $a_n$, where: $a_0 = a,$ $a_1=b,$ $a_{n+2}=6a_{n+1}-9a_n$ By looking at $a_2, a_3, a_4$... I did manage to see some formula, but I don't think this is the right ...
5
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4answers
1k views

Solving Coupled Eigenvalue equations

I wish to solve the following set of coupled eigenvalue equations. How should I do it? For real matrices $A$,$B$,$D$ and vectors $x \in R^m$, $y \in R^n$ $$ A x + B y = \lambda x $$ $$ B^T x + D y ...
5
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2answers
340 views

Is there a name for the matrix $X(X^tX)^{-1}X^{t}$?

In my work, I have repeatedly stumbled across the matrix (with a generic matrix $X$ of dimensions $m\times n$ with $m>n$ given) $\Lambda=X(X^tX)^{-1}X^{t}$. It can be characterized by the following:...
4
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2answers
30k views

Checking axioms of Vector Spaces

Currently I am studying a section from my book on vector spaces. I'm having issues in understanding how I am supposed to prove some of the questions in the Exercises section, such as: In each of ...
2
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2answers
339 views

Faithful extension (Is the image of a polynomial map an algebraic set? )

Let $k[x_1,\ldots,x_n]$ be a polynomial ring, $k$ be an algebraically closed field. Suppose $k[T_1,\ldots,T_m]$ is a finitely generated $k$-subalgebra such that for any proper ideal $I$ of $k[T_1,\...
2
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1answer
1k views

Linear regression or average percent change?

I have 12 (1 year's worth) values and I want to determine the trend and represent the trend as a positive or negative figure. What is the difference in using linear regression versus the monthly ...
7
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2answers
2k views

equivalence between matrix multiplication and matrix inversion

I am a bit confused with this wikipedia article, hoping someone can clarify it. Looking at the Strassen algorithm page it is clear that this is an algorithm for reducing multiplication operations. ...
4
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5answers
653 views

Help a newbie understand Linear Algebraic terms

I am taking a class in Algebra but I am having a problem grasping exactly what it is I am being asked to do -- I think I am having a problem with the vocabulary being used. I have a couple of ...
2
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1answer
369 views

Matrix column permutation under constraint

Apologize if you've read my question on Mathoverflow, I'm very curious about whether there's an answer to this. In coding theory, there are parity-check codes whose parity-check matrices H are ...
1
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1answer
941 views

computing Moore-Penrose pseudoinverse when SVD computation does not converge

I am writing a routine to return the Moore-Penrose inverse of a rectangular matrix. Currently am computing the Moore-Penrose inverse using SVD, i.e., if the SVD is given by $A = \sum_{i=1}^r \...
20
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1answer
2k views

Prove that a matrix is invertible [duplicate]

Show that the matrix $A = \begin{bmatrix} 1 & \frac{1}{2} & \ldots & \frac{1}{n}\\ \frac{1}{2} & \frac{1}{3} & \ldots & \frac{1}{n+1}\\ \vdots & \vdots & & \vdots \...
4
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2answers
238 views

rank of a matrix

First of all I am sorry because I have asked similar kind of question a few days ago.But I still have problem with row reductions when there are letters in a matrix.The question is asking the value of ...
1
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1answer
324 views

Blockwise inversion case when $\textbf{D} - \textbf{C}\textbf{A}^{-1}\textbf{B}$ is singular

What means in blockwise matrix inversion when $\textbf{D} - \textbf{C}\textbf{A}^{-1}\textbf{B}$ is singular but $\textbf{A}$ is not? is that necessary and sufficient for the whole composed matrix be ...
2
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0answers
228 views

multi-index linear equation without computing explicit inverse

This is a computational-related question, i couldn't find a better SE forum for this. I have a set of equations for $ \textbf{Z}$ 3-index quantities of the form $$ S_{mi} S_{nj} Z^{k}_{ij} S_{kp} = ...
5
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4answers
31k views

Conditions for vectors to span a vector space

I have a basic doubt. Can we say that a set of vectors span the entire vector space iff they are linearly independent ? Do they need to satisfy any other property ?
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3answers
3k views

Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?

I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these. It seems to be a well known result that when you take the eigenvectors of $A$ as a basis and ...
3
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0answers
284 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 $(v_i,v_j^*)=\delta_{...
4
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0answers
213 views

image of symmetric matrices under representation of $GL_2(\mathbb{R})$

Let $W$ be a real vector space of dimension $2$ and let $\rho_k:GL_2(\mathbb{R}) \to GL(\mathbf{S}^kW)$ be the standard representation of $GL_2(\mathbb{R})$. Since $\rho_k$ is polynomial, it naturally ...
1
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2answers
196 views

Characteristic polynomial and $p$-adic valuation

Suppose I had a linear operator $L$ whose characteristic polynomial was $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n-1}x + a_{n}$. Furthermore, I also know that the eigenvalues of $L$ have $p$-adic ...
1
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1answer
94 views

Condition to be able to decompose a finite-dimensional real vector space V into kernel and image of a linear map T from V to V

(I will phrase the question in terms of $\mathbb{R}^n$) Is the following statement a standard well-known linear algebra fact that I can quote without proving? (Perhaps more importantly, is it even ...
3
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2answers
393 views

Commutative Matrix Problem

What are values of $a,b,c$ such that matrix $A$ and $B$ commutes, i.e. $AB=BA$, where $A$ and $B$ are (3,3) matrix such that $A=\pmatrix{a_0 & b_0 &c_0\\ a_1 & b_1 & c_1\\ a_2 & ...
1
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1answer
123 views

linear algebra - equations

Determine real number(s)for $a,b$ such that the system has no solution, has a unique solution, and has more than one solution: $$\begin{align*} x-2y+az-t&=1\\ -x+y-z+t&=-1\\ (a+1)y-a^2z+...
3
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1answer
2k views

Projecting a nonnegative vector onto the simplex

Given an elementwise nonnegative vector $y$, I'd like to find the projection of $y$ onto the simplex $S: \{ (x_1, \ldots, x_n) ~|~ \sum_{i=1}^n x_i=1, x_i \geq 0 \mbox{ for all } i \}$. Is there a ...
0
votes
1answer
102 views

Describing fans from hyperplanes

If $H_1, ..., H_n$ are hyperplanes in $\mathbb{R}^m$ such that the complement of the union $\cup_i H_i$ is the interior of a complete polyhedral fan, then how does one determine ray generators for ...
0
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1answer
464 views

How to partial differential of a trace of matrix form?

How to prove that $$\frac{\partial \mathrm{Trace}(ABA^T)}{\partial A}= AB+AB^T$$ if $A$,$B$ are square matrices. Can I write the following: Let $C=BA^T$. Then, $$\frac {\partial \mathrm{Trace}(...
11
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5answers
17k views

Proving that two systems of linear equations are equivalent if they have the same solutions

I've just begun to work learn Linear Algebra on my own through Hoffman and Kunze's book and the first problem set already has a question that I can't solve: Prove that if two homogeneous systems of ...
4
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2answers
752 views

Programming Logic: - Splitting up Tasks Between Threads

I asked this question at stackoverflow and instead of addressing the math required in the problem, they wanted to talk about why setting up 5 threads is no good, or question my intentions. I just want ...