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 ...

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Solving for A in the system Ax = 0

Consider the system of linear equations $A x = 0$ where $A$ is a $K \times M$ matrix of reals and $x$ is an $M \times 1$ vector of reals. The matrix $A$ is unknown but we can generate $x$s that ...
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60 views

How to find a hilbert basis of a given subspace considering a given inner product

Let $X$ be the space of continuous functions on $[-1;1]$ to $\mathbb{R}$ with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let $U$ be a subspace of $X$ with $U := ...
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98 views

Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...
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47 views

Regularity of a tridiagonal matrix

I want to know, in which cases i can say that a tridiagonal matrix is regular, i.e. it has an inverse? I know that it is not strong diagonal dominant and in think i have to use that $A$ is regular if ...
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57 views

Why is the dimension of the column space of a matrix $A$ equal to the dimension of the row space of $A$?

The proof I have in the lecture notes is as follows: $$\begin{align*} \dim(Im(A)) &= \#\text{ number of vectors in basis of } Im(A) \\ & =\#\text{ leading 1's in echelon matrix of }A \\ & ...
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416 views

Proof that if a matrix is invertible, its rank is maximum

I have to prove that if a square matrix $A \in \mathfrak{M}_n (\mathbb{K})$ is invertible, then $rg(A) = n$. The thing is I cannot use vector spaces, subspaces, etc... to prove this, only matrix ...
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33 views

Replacing a vector in a basis for a subspace of $R^N$ (numerically)

Say I have some basis ${u_1, u_2, .., u_k}$ for a subspace $ U \subset R^N$ (where $k$ and $N$ are typically very large). I have another vector $v \in U$. How do I (efficiently) find an $i \in ...
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31 views

Exterior product of vectors in $\mathbb{R}^4$ with integer coefficients.

Let $a, b, c, d$ be vectors with integer coordinates in $\mathbb{R}^4$ such that $k a \wedge b = c \wedge d$ for some integer $k$ and $a \wedge b \neq l v$ for any $v \in \bigwedge^2 (\mathbb{R}^4)$ ...
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23 views

Show C is not 1-error correcting by using Slepian decoding

Let C $\subseteq$ $ \mathbb{Z}_2^5$be a linear code with generator matrix $$G=\begin{bmatrix}1 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 1\\0 & 0 & 1 & 0 & ...
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20 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
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32 views

Nonlinear animation algorithm

I'm a programmer working on an animation algorithm. Heuristically, it starts out like this: ...
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110 views

What is common between difference operators and recurent relations?

They say that solutions of recurrence relations are combinations of exponential functions, the series like [1 a a^2 a^3 and etc]. I know that the difference operators have a matrix like ...
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43 views

Showing linearity

Is this function linear? $ \mathbb{R}^{\mathbb{N}} \to \mathbb{R}, (x_n)_{n \in \mathbb{N}} \to \lim_{n\to \infty} x_n$ Well.. Usually I have to show $L1: F(x+y) = F(x) + F(y)$ $L2: k\cdot F(x) = ...
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32 views

What may I infer from knowing that some data set's covariance matrix is singular?

I know what a matrix being singular means. I know what a covariance matrix is. What I'd like to know, though, is what I can infer about a data set if I know that its covariance matrix is singular. ...
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94 views

Prove that a linear operator is indecomposable

Let $V$ be a fi nite-dimensional vector space over $F$, and let $T: V \rightarrow V$ be a linear operator. Prove that $T$ is indecomposable if and only if there is a unique maximal T-invariant proper ...
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55 views

Derivative of a tensor

I have a rank-2 tensor given by $$ P_{\alpha \beta} = p\delta_{\alpha \beta} + (u_1^2, u_1u_2 ; u_1u_2, u_2^2) $$ whose derivative with respect to $x_{\alpha}$ I would like to find. According to my ...
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51 views

Norm of a matrix exponential

Can any one prove the following inequality $$||e^{Pt}||\leq e^{t\alpha{(P)}}\sum_{k=0}^{r-1}\frac{(||P||\sqrt{r}\,t)^k}{k!}$$, where $r$ is the order of the matrix $P$ and $\alpha(P)$ be the maximum ...
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74 views

Does the derivative of the largest eigenvalue and its associated eigenvector exist?

Let $\boldsymbol{A}(t):\mathbb{R} \rightarrow \mathbb{C}^{n\times n}$ be a rank-deficient Hermitian function-valued matrix (its entries are analytic functions). Furthermore let $\lambda_1(t)$ (which ...
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40 views

find a the minimal polynomial ofa 6*6 matrix

Question: find minimal polynomial of $\begin {bmatrix} 1 & -4& 2 & 8& -6&-2 \\ -2&-3&4&10&-6&-2 \\-5&-2&6&9&-3&-3 \\ ...
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51 views

Transition Matrices

I am given a set of vectors $S$ and that they are a basis of a vector space $V$. I am told that there is also another basis {$v_1,v_2,v_3,v_4$} for $V$. I need to find a transition matrix $A$ from the ...
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74 views

Find T cyclic operator that has exactly N distinct T-invariant subspaces

Let $T$ be a cyclic operator on $R^3$, and let $N$ be the number of distinct T-invariant subspaces. Prove that either $N$ = 4 or $N$ = 6 or $N$ = 8. For each possible value of $N$, give (with proof) ...
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186 views

Solving linear system of equations using Successive Over-Relaxation

I was solving a system of linear equations with SOR. I used different values of relaxation factor (w) for the different runs. I found that for all w > w' (1 < w' < 2), the error is the result ...
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122 views

Proof by contradiction. Which statement has to be shown to be false?

I want to prove the following statement: Show that if $B=(b_1,....,b_n)$ is a basis of a vector space V, then there is no list of vectors of length $n-1$ that spans V. I would like to prove this by ...
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50 views

Vanishing criterion of pure wedges

Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
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78 views

Deficienting rank of a matrix

Dear friends Let ‎$\bf{C}‎$‎ be a ‎$m \times n‎$ ‎matrix, where its elements are drawn randomly from a continious distribution, and its rank is ‎$\min (m, n)$‎ with probability one. For ‎decreasing ...
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157 views

Orthogonal Procrustes Problem

The classical orthogonal Procrustes problem concerns finding the matrix $\Omega$ which minimizes $||A\Omega-B||_{F}$ subject to $\Omega'\Omega=I$, with A and B known matrices. Let A be the identity. I ...
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54 views

What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
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155 views

Method of determining dimensions from photographs of multiple angles and degrees of perspective/parallax for a math newbie

I have a project that begins with some 300+ reference photos of a scale model. The only measurements I am certain of are the overall length, and the linear length of one element of one part of the ...
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76 views

Some problems with a proof of the Farkas Lemma

The following is a proof of the Farkas Lemma that is creating me quite some problems. [I presented the all proof simply to point out the notation used by the author.] My problem is with the last part ...
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53 views

Why the pivots and relative to the eigenvalue in symmetric matrix?

In the book, it said, there a quick fast way to test whether the eigenvalue are all positive or not. Just check the pivot of the symmetric matrix, if x no. of positive pivot, it would have x no.of ...
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80 views

Invertibility of Augmented matrix [ A | b ] & Solutions of Ax = b [GStrang, P182, 3.5.39]

Suppose $A$ is $5$ by $4$ with rank $4.$ Show that $\mathbf{Ax = b}$ has no solution when the $5$ by $5$ matrix $\begin{bmatrix} \mathbf{A} & \mathbf{b} \\ \end{bmatrix}$ is invertible. Show ...
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Is computing the inverse of a complex matrix different than for a real-valued matrix?

I’m tasked with writing an algorithm to find the eigenvalues and eigenvectors of $A \in \mathbb{R}^{n\times n}$, with $A$ being upper triangular. The process is quite simple with the eigenvalues just ...
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55 views

Groups Of Rigid Motion

Is the group of rigid motions ${G}$, the largest group of linear transformations in $F^n$ with composition as the product such that $<Av,Aw>=<v,w>$ where $A \in {G}$ and $v,w \in F^n$ and ...
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28 views

If X and Y are sequential tangents to a group G at the identity matrix, show that X+Y is also.

If X and Y are sequential tangents to a group G at the identity matrix, show that X+Y is also. Definition: X is a sequential tangent vector to G at the identity matrix I if there is a sequence ...
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122 views

(When) is this matrix positive definite?

I have a symmetric $n \times n$ matrix (say, $M$) with $[i,j]$ element \begin{equation} M_{[i,j]} = \int_{\mathbb{R}} [p_i(z)-g(z)][p_j(z)-g(z)]~dz, \end{equation} where $p_i(\cdot), p_j(\cdot),$ ...
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76 views

Gauss-seidel and implicit method

I have a matrix $\mathbf{X}$ and I want to apply a function $f_{ij}$ to each entry of it, until convergence is satisfied. If a value is known in this matrix, then the $f_{ij}$ at this point may be the ...
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43 views

Constraints on eigenvalues due to the form (symmetry) of Matrix

So I have to deal with a square matrix with complex elements $A$ given by - $$A_{ij}(k) = \underbrace{\frac{1}{2}(c_{ij}+c_ji)}_{\text{symmetric under interchange of $i$ and $j$}} \exp(Ik(i-j))$$ ...
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45 views

Dual representation matrix “recycling”

Imagine we have $V$, a finite dimensional vector space endowed with an inner product and its dual space $V^*$. We have also a matrix Lie algebra and a representation of it, $\pi$, that acts on $V$. ...
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73 views

Intuition - Zero Vector, Its Existence in Any Set, and Linear Dependence [GStrang, P169]

$I.$ The zero vector is linearly dependent. $II.$ Any set containing the zero vector must be linearly dependent. I only apprehend the truths of I and II above from the definition of linear ...
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237 views

Consistency of linear system of equations

Let $Ax = B$ be a system of linear equations, where $A$ is an $n \times n$ matrix and $x, B$ are $n \times 1$ vectors. Yesterday in class our teacher said that if det$A = 0$ and Adj$A.B = O$ (where ...
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106 views

Calculating the quadratic form of a vector and Kronecker product

I am trying to calculate the quadratic form $A = BQB^T$ where $Q = R \otimes S$. $R$ and $S$ are $[r,r]$ and $[s,s]$ square matrices, and $B$ is a $r \times s$ vector. Can I express $A$ in terms of ...
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133 views

How to compute this recursion in linear time?

Can the following iterative update on a $n$-element vector $\mathbf{x}_t$ be computed in $O(n)$ computations? \begin{align*} \mathbf{x}_{t+1} & = a_t\mathbf{y}_t + \mathbf{A}_t \mathbf{x}_t \,,\\ ...
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45 views

Rate of Change with respect Q

$M = 2/3 * \log10(E/0.007)$ I need to calculate the rate of change of $M$ with respect to $E$ if $E = 60,000$ and previously $M = 7.7$ It's been a while since I did this and it's for a friend so ...
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214 views

Orthogonal complement of orthogonal complement $w^{\perp\perp}$

For any subspace W of a vector space V, prove that $$ W + V^\perp = W^{\perp\perp}. $$ I am having difficulty showing: $$ W^{\perp\perp} \subseteq W + V^\perp . $$ Ideas?
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60 views

Orthogonal Procrustes Variant

(author note: this question was also asked on mathoverflow). The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both ...
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27 views

How can I find $f^{ad}$ along with $\operatorname{Im} f^{ad}$ and $\operatorname{Null} f^{ad}$?

Let $ f \in \mathbb R_{\leq4}[t]$ such that $f(p)=f \bigg( \sum\limits_{k=0}^4 a_k t^k\bigg):=f \bigg( \sum\limits_{k=1}^4 k a_k t^{k-1}\bigg)$ and define $\langle p,q \rangle := \int\limits_{-1}^1 ...
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77 views

Eigenvectors of a random non-Hermitian symmetric tridiagonal Matrix

Are eigenvectors of a random non-Hermitian symmetric tridiagonal Matrix (with complex components) localized?
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36 views

Hessian of element of symmetric square root

This is related to a previous question I had. I'm trying to find the Hessian or second differential for an element of a symmetric square root matrix (found by SVD or spectral decomposition). So, if ...
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154 views

Partial Derivative of a Scalar-product Resulting from Vector Multiplication

I am trying to differentiate the function below, but I am running into problems due to the point-wise multiplication with the matrix. $f(x,y,A) = (x^{T}y) \cdot A$ Where $\cdot$ denotes point-wise ...
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47 views

Schwarz Inequality Proof: The Mystery of the Disappearing Vector

I've been scratching my head over this for about 45 minutes now, and I have no idea where why the $w$ in this proof disappeared. $\left| \left \langle u,v \right \rangle \right| \le ||u|| ||v|| $ ...