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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Jordan canonical form in Lang's Algebra

In Lang's algebra on pp.559, he writes of the nilpotent part of a matrix $M$: "We observe also that the only case when the matrix $N$ is $0$ is when all the roots of the minimal polynomial have ...
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194 views
+100

Positivity of the alternating sum associated to at most five subspaces

Let $V_1 , V_2 , \dots , V_n $ be proper subspaces of $ \mathbb{C}^m$ and let $$\alpha = \sum_{r=1}^n (-1)^{r+1} \sum_{ \ i_1 < i_2 < \cdots < i_r } dim(V_{i_1} \cap \cdots \cap V_{i_r})$$ ...
2
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1answer
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Looking for a particular algebraic mapping from one Boolean matrix to another

Consider the following Boolean matrix: \begin{align} X&=\begin{bmatrix} 1&1&1&1&1&1&1&1\\ 1&1&1&1&0&0&0&0\\ ...
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1answer
525 views

Reconstructing an optimal Simplex tableau from an optimal solution

I have here a bounded LP with infinite optimal solutions: ...
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1answer
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Existence of a block upper triangular form matrix representation for a linear operator

Let $T:V \rightarrow V$ be a linear operator on a finite dimensional vector space over $F$. Let $W \subset V$ be a subspace which is $T$-invariant. Show that there exists an ordered basis ...
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Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
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263 views

What is that thing that keeps showing in papers on different fields?

A few months ago, when I was studying strategies for the evaluation of functional programs, I found that the optimal algorithm uses something called Interaction Combinators, a graph system based on a ...
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Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
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$k[x_1, \dots, x_n]$ free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism.

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
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368 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $$V= \begin{bmatrix} ...
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Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
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119 views

Integrating a matrix function involving a determinant and exponential trace

I am trying to find the normalizing constant for a probability distribution and ran into a difficult integral. When $X$ is an $p \times k$ matrix, $a>0,$ and $g>0,$ how can I compute $$\int ...
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188 views

On the maximum number of polynomials in a certain subspace

I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that. Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with ...
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551 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
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What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices ?

Given a set of points $x_1,x_2,...,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m$ x $m$ Euclidean Distance Matrix $D$ where $D_{ij}={||x_i-x_j||}^2$. We know a little bit about these ...
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What does the space of non-diagonalizable matrices look like?

Let $k$ be a field (I would be happy working entirely over $\mathbb C$). Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices ...
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198 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
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191 views

Given an irreducible representation, is there a *unique* unitary representation that it is equivalent to?

I might need help here in understanding my own question in places and please don't hesitate in asking for edits and clarifications. Background: A representation $\rho$ of a finite group $G$ is a ...
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Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
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Linear transformation as dot product

Prove that to every $A\in L(\mathbb{R}^n,\mathbb{R}^1)$ corresponds a unique $\mathbf{y}\in \mathbb{R}^n$ such that $A\mathbf{x}=\mathbf{x}\cdot \mathbf{y}$. Prove also that $\Vert A ...
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58 views

cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
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52 views

Orthogonal projectors on non-orthogonal subspaces

It is a well known fact that if(f) $V,W$ are orthogonal subspaces of a Hilbert space $H$, then their orthogonal projectors satisfy: $$ P_{VW} = P_V + P_W, $$ where $P_{VW}$ is the projector on $V+W$. ...
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279 views

Centralizer of a Matrix over a Finite Field

Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$. For a matrix $A\in M_n(\mathbb F)$ what is the cardinality of $C_{ M_n(\mathbb ...
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Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
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29 views

About a particular definition of “tensor”

I came across this quiet new to me way of defining "tensors", That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n ...
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23 views

Awkwardly formed linear spaces exercise

I came across such an exercise: Let $V$ be a linear space over $K$ such that $\dim V = n$. Show that for any $\alpha_1, \alpha_2, \dots, \alpha_m$ with $ m > n + 1$ there exist $a_1, \dots, ...
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46 views

Is $\{(x,y,z) \in \mathbb{R}^3 : x^2+3y^2+12z^2=0\}$ a vector space?

Is $\{(x,y,z) \in \mathbb{R}^3 :x^2+3y^2+12z^2 = 0\}$ a vector space? My inclination is that the only real solution to $x^2+3y^2+12z^2=0$ is $(0,0,0)$, which is the trivial subspace of ...
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25 views

What is the maximum value of coefficient $f_v$ with the constraInt that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
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Set of conjugate vectors that span both Krylov space

If $P$ contains a set of conjugate vectors that span Krylov space of matrix $A$, i.e. $\mathcal{K}(A, x)$, and also $P$ span Krylov space of matrix $\mathcal{K}(B, x)$, is it true that the diagonal ...
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60 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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39 views

Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
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Extending the trace inner product to all matrix (real) inner products

In ${\bf R}^{n\times p}$ we have the trace inner product given by $$\langle A, B\rangle=\text{tr}(A^TB)$$ which can be interpreted as the Euclidean inner product on ${\bf R}^{np}$. All inner ...
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22 views

The sign of the complementary minor of a Matrix

I am stuck on this problem which looks rather basic : Let $A$ be any matrix .Prove that the sign of a minor $A[I,J]$ and its complementary minor $A[I',J']$ are the same...... Basically I am looking at ...
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40 views

A question about adjoint matrices

Let $T:V \to V $ be a linear map on complex vector space $V$ which is equipped with complex inner product $ <. , .> $ we know there exists a unique linear operator $T^* : V \to V $ such that ...
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Characters of Representations, Composition Series and Tensor Products

Let $(\pi, V)$ be a finite-dimensional representation of $G$. Prove the following: Suppose that $(\pi, V)$ has as a composition series $\{0\} \subset V_{1} \subset \dots \subset V_{r}=V$ with the ...
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If $T$ is an orthogonally diagonalizable linear operator in an inner product space, show that $T^*$ is also orthogonally diagonalizable.

Suppose $V$ is an inner product space and $T$ is a linear operator that is orthogonally diagonalizable. Show that $T^*$ is also orthogonally diagonalizable. Here, $T^*$ denotes the adjoint ...
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13 views

Spectral norm of lower triangular perturbation

Suppose $A\in R^{n×n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. \begin{equation} A=I+L \end{equation} All diagonal entries of $L$ are equal to $0$, so that, ...
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15 views

Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...
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$M_n$ is the subspace of all square matrices with trace $0$, what is the dimension of $M_n$?

There is an older post with many explanations of a more specific and less general case of a $4$ by $4$ Find the dimension of the space of $4\times 4$ real matrices with zero trace I didn't quite ...
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Linear Algebra - Transition matrices

Question I have some methodological questions with this exercise: 1. You are given that the transition matric $P_{\mathcal C,\mathcal B}$ from a basis $\mathcal B=\{b_1,\ b_2,\ b_3\}$ to a basis ...
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Why is the standard inner product on F^n equal to this?

In the textbook that I'm using the standard inner product is defined as $$\langle x,y\rangle = \sum_{i=1}^{n}a_{i}\overline{b_{i}}$$ where $x=(a_{1}, a_{2}, {...}, a_{n})$ and $y=(b_{1}, b_{2}, ...
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PGL_n transformation fixing a set of n+2 points in general position must be identity

I was wondering if anybody can think of a slick and short argument to show that any $PGL_n(k)$ transformation $A$ that fixes a set of $n+2$ points $p_1, p_2,\dots, p_{n+2}$ in general position must be ...
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24 views

Twisted centralizer

Let $F$ denote a finite field and $A$ a square matrix with coefficients in $F$. The set of all matrices $B$ such that $BA=AB$ is called the centraliser of $A$. Now consider the set $C(a,A)$ of all ...
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26 views

Find the rotation angles of a 2-D rotation matrix between two vectors

I am trying to solve the following to find $\theta$. I was given two vectors $\begin{bmatrix}-4.95 \\ -.7\end{bmatrix}$ and $\begin{bmatrix}3 \\ 4 \end{bmatrix}$ and asked to compute the rotation ...
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Can I use the inverse map to show that the map is bijective (isomorph)?

I want to show that a map f: V to V is isomorph. I found the inverse map of it and wanna know if it will do that I write the inverse function down and show that it's a homomorphism. Or have I to show ...
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0answers
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Help needed with linear algebra computation

I do not see what is happening here. $$(F^TF + \begin{bmatrix}V_1 & V_2\end{bmatrix}\begin{bmatrix}\gamma &0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}V_1^T \\ V_2^T\end{bmatrix})^{-1}F^Ty = ...
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0answers
25 views

Writing a Rotation Matrix About an Angle

I am asked to find a rotation matrix $R_O$ of an angle $O$ about axis $u\in R^3$, with $u$ having length of 1. I've looked up this concept on the web but I have no idea where to get started...could ...
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Let P4 be the vector space of polynomials of degree at most 4. For the following map decide if it is an isomorphism?

How can i describe this operation as an isomorphism or not ? $p(x)=p_0+(p_1)x+(p_2)x^2+(p_3)x^3+(p_4)x^4\longmapsto q(x)=p_1+(p_2)x+(p_3)x^2+(p_4)x^3+(p_0)x^4$
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$S(0,\varepsilon ) \Rightarrow F + S(0,\varepsilon ) = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$

Let $F \subseteq {\rm{C}}$ and $S = \left\{ {x \in C:\left\| x \right\| \le \varepsilon } \right\}$. Why does $F + S = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$? (where ...