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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Find the number of distinct real values of $c$ such that $A^2x=cAx$

Let $$A= \begin{pmatrix} 5 & -3 & 0 \\ -3 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}$$ and $c$ be a real no. such that $A^2x=cAx$ for some non-zero vector $x$. Then the number of ...
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23 views

Weight spaces of a irreducible representation of $\mathfrak{gl}(n, \mathbb{C})$.

Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be ...
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Find a symmetric matrix B that makes ABC symmetric, A,C known

I have two known matricies $\bf{A} \in \mathbb{R}_{nxm} $, $\bf{C} \in \mathbb{R}_{mxn}$ with $m>n$. I'm trying to find a $\bf{B} \in \mathbb{R}_{mxm}$ that is symmetric and makes $\bf{ABC}$ ...
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31 views

Gram-Schmidt in characteristic two?

I was helping someone work on a computing problem with bit vectors that reduced to finding a basis knowing a spanning set, and realized quickly that the Gram-Schmidt process does not work as expected ...
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A matrix representation for the inverse matrix.

I have the next problem from the textbook: "Methods of Algebriac Geometry in Control Theory by Peter Falb". Problem 1: Show that if $A$ is an $n\times n$ matrix, then $(zI-A)^{-1} = \sum_{j=1}^n ...
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Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
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Angle between two vectors in $\mathbb{R}^n$?

I know that in $\mathbb{R}^2$, the dot product of two vectors $\bf a$ and $\bf b$ is given by $$\mathbf {a} \cdot \mathbf {b} = \mathbf{|a|\,|b|}\cos(\theta), \tag 1$$ where $ | \cdot |$ denotes the ...
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49 views

Proof that Determinant is Scale Factor

I've seen a lot of supposed properties of linear transformations that're never proven -- just often repeated. These include: The determinant is the scale factor between the volume of region in your ...
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38 views

Finding a summarizing vector for average angle calculation

Let $L$ and $R$ be two bags of positive vectors such that all vectors have length $k$. Define the distance $d_{avg}$ between the bags as the average pairwise angle between the vectors. Is is possible ...
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51 views

Basis of $\mathbb{F}[[x]]$ over $\mathbb{F}$ without AC

Does the ring of formal power series $\mathbb{F}[[x]]$ as a vector space over $\mathbb{F}$ admit a basis without assuming the Axiom of choice, at least in some special cases of $\mathbb{F}$? I'm ...
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23 views

Convergence of Arnoldi method

I would like to compute the largest real eigenvalue of a matrix in the following form: $$\begin{bmatrix} 0 & I_n \\ P & Q \end{bmatrix},$$ where $I_n$ is the $n \times n$ identity matrix, $P$, ...
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33 views

Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
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Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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Number of solutions of an equation

Fix a vector $x\in\{0,1\}^n$, and let $a$ be a random vector in $\mathbb{Z}^n_q$ for some prime $q$. Consider $y=ax$, and $S=\{x'\mid ax'=y\}$. I want to compute the probability that $\lvert S ...
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17 views

Optimal Matching Distance

I'm stuck on problem II.5.9 from Bhatia's Matrix Analysis. The problem is as follows: Let $\{\lambda_1,\dots,\lambda_n\},\{\mu_1,\dots,\mu_n\}$ by two $n$-tuples of complex numbers. Let $$ ...
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44 views

Decompose a vector space into invariant subspaces?

Consider the following proposition: Suppose $V$ is a finite dimensional vector space over a field $F$, and $K/F$ is a finite Galois extension with Galois group $G$. If $V$ has a $(K,K)$ bimodule ...
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Does this have a name?

While messing around, I seem to have stumbled upon an interesting family of matrices: $$\mathbb{S} = \bigg\lbrace A\in\mathbb{M}_{n\times n}(\mathbb{R}) : A^{T}A=AA^{T}=\frac{1}{2} (A + ...
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Forming the graph $G$ from elements of the cut and cycle space, using a weird hint

I'm working through a set of lecture notes on my own, and since there is no class, there are no immediate faculty members available to ask questions to. I've managed to finish most exercises quite ...
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53 views

Relation between $y=Ax_1$ and $y=WAx_2$

I have a question. Is there any relation between the following linear equations? $$y=Ax_1 \ \ \text{ and} \ \ y=WAx_2$$ W is diagonal square invertable matrix, A is an mxn matrix with $n>m$. I ...
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47 views

Eigenvalues of adjugate matrix of a singular matrix

Given a singular matrix $A$, find the eigenvalues of the adjugate matrix of $A$. The same question with $A$ being invertible is trivial since $A\operatorname{adj}A=(\operatorname{adj}A)A=(\det A) ...
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34 views

Iteratively transform vectors to/from the basis of eigenvectors

I have a $n \times n$ banded symmetric matrix $A$ with bandwidth $k \ll n$, so it is quite sparse. The matrix is diagonalizable, $A = V^T \Lambda V$. Generally, $A$ is not positive definite. I need ...
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If $A$ is normal then $A^*=P(A)$ for some polynomial $P$.

Since $A$ is normal it has a diagonal matrix with respect to some orthonormal basis, and the same is true of $A^*$ using the same unitary matrix. We also know that if $\lambda$ is an eigenvalue of $A$ ...
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115 views

Relationship between eigenvalues of two related, Euclidean distance matrices

If $X=\{x_1,\ldots,x_N\}$ is a set of points in $\mathbb{R}^n$ then one can generate a Euclidean distance matrix $D = [d_{ij}]$ where $d_{ij}=\Vert x_i-x_j\Vert_2^2$ is the square of the Euclidean ...
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59 views

Mapping vector spaces over two different fields?

I was having linear algebra class and we have been discussing about a possible group homomorphism that might allow mapping between two vector spaces over two different fields This is also an ...
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38 views

Find all $ p \ge 1 $ for which the Hölder norm $\|\cdot\|_p $ is generated by a scalar product.

Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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Is this the correct solution involving vector subspaces and basis?

I need to find the basis and hence dimension of a subspace of $\mathbb{R^3}$. 1) $$U=\{(x,y,z):x=2y\}$$ Solution: We have $x=2y \iff y=\frac{x}{2}$ therefore we can write all elements in $U$ as the ...
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74 views

Matrix which is not similar to it's transposed

Let $V$ be vector space over a field $\mathbb{k}$. I can prove that any matrix is similar to its matrix transpose if $\mathbb{k}$ is an infinite field, but is this still true when $\Bbb k$ is finite? ...
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47 views

Matrix equation $A=PBP^{-1}$

Suppose $P$ is invertible and $A=PBP^{-1}$ . Solve for $B$ in terms of $A$. My attempt: I just left multiplied the equation by $P^{-1}$ and right multiplied it by $P$ so that I got $B=P^{-1}AP$. Is ...
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54 views

Which of the following subsets of $\mathbb C$ is a field?

I'm not entirely sure that I understand the concept of fields fully so I'll give you the question and then I'll let you pick my brain and tell me if my logic is correct. Please note: I'm not just ...
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82 views

Convergence of integers by transformations

Let $x=(a,b)$, where $a,b$ are in $N$ Now we have the transformations: $$T_1(x) = (ka, b+1)$$ $$T_2(x) = (b,a)$$ where $k$ is in $N$. Where the order of choosing a transformation is not fixed. ...
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48 views

When does $\| \Pi \|_1 = 1$ where $\Pi$ is a projection.

By projection I mean any matrix such that $\Pi = \Pi^2$. It is well known that all projections can be written as $\Pi = A(B^\top A)^{-1}B^\top$ for some $A,B$. Characterize the class of projections ...
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Eigenvalues of Overlapping block diagonal matrices

I look for eigenvalues of general overlapping block diagonal matrices. e.g. $$\left[ \begin{matrix} 1 & 4 & 0 & 0 & 0 & 0\\ 4 & 2 & 3 & 2 & 0 & 0\\ 0 & 3 ...
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42 views

Hadamard products VS Matrix product

Some notation before introducing the question. We discretize an interval $[0,L]$ in uniform subintervals, each with length $\Delta x$ and we assume that vectors living over such a grid satisfies ...
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43 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
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The relation between $\inf_{R\in \mathsf{U}_n} \left\Vert A - BR\right\Vert^2_F$ and $\left\Vert AA^*-BB^*\right\Vert$

Suppose that $A$ and $B$ are two arbitrary $m\times n$ matrices with $m>n$. Let $\mathsf{U}_n$ denote the set of $n\times n$ unitary matrices. I'd like find positive constants $c_1$ and $c_2$ such ...
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quadratic form from nxn matrices to reals ( Tr(A^2) ). I need to find it's signature and rank.

Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm ...
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Multivariate polynomials with prescribed zeros

Let $P_k\subsetneq\Bbb R[x_1,\dots,x_n]$ be the set of degree $k$ multivariate real polynomials. Pick a subset $S$ of $\{-1,+1\}^n$ of size $|S|<\sum_{i=0}^k\binom{n}{i}$. We seek a polynomial ...
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41 views

Interpretation of a 3 Variable System of Equations

I'm a high school student, and, of course, this week is finals week. For my Algebra 2 semester final, we have been permitted to take the test home and collaborate with others. This final can be viewed ...
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51 views

Finding $n$ scalars such that $\det{(cI-A)}=0$ without eigenvalues

My problem is this Let $A$ be an $n\times n$ matrix over $\mathbb{F}$. Prove there are at most $n$ distinct scalars $c\in\mathbb{F}$ such that $\det{(cI-A)}=0.$ I know that the determinant is ...
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How do $\Bbb{R}$ and $\Bbb{R}^2$ have the same dimension over $\Bbb{Q}$ as vector spaces?

How do $\Bbb{R}$ and $\Bbb{R}^2$ have the same dimension over $\Bbb{Q}$ as vector spaces? Having the same dimension would imply that both have the same number of basis elements. How do I prove that ...
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Simultaneously diagonalizable without distinct eigenvalues

It is a well known result that if $u$ and $v$ are two diagonalizable endomorphisms of a $\mathbb{C}$ finite-dimensional linear space $E$, if $u$ (or $v$) has distinct eigenvalues and if $u$ and $v$ ...
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If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$ where $S$ is a invertible matrix and $B$ has the form $B = \left( \begin{array}{ccc} 0 & a_1 & 0 & ...
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Shortlist of problems in linear algebra

A while ago I remember seeing a very nice shortlist of problems in linear algebra. It was a list of about 40-50 problems. The idea was that if you solve them, you learn linear algebra very well and ...
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Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
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Finding the new basis for a diagonalized quadratic form

I have been given the quadratic form $$A(x,x) = 2x^2-\frac{1}{2}y^2-2xy-4xz$$ and been asked to diagonalize it, find the change of basis matrix, and find the new basis in which A is diagonalized. I ...
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How to derive the following from Azuma's inequality?

This is claimed in Proposition 1 in the paper http://arxiv.org/abs/1409.6110 Let $A$ be a $n \times d$ matrix. $A$ can have only $K$ different types of rows i.e. rows of $A$ are chosen from a set of ...
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37 views

the exact graph of the general solution for $x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$

i need someone to give me exact graph of the general solution for $$x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$$ i solved it manually , the general solution is like this ...
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58 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
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118 views

Jordan normal form theorem proof question

Theorem: Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan ...
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59 views

Find the projection of any vector onto the linear span and the normal from any vector to that span

Show that the vectors $u_1 = (1/9,4/9,8/9), u_2=(8/9,-4/9,1/9), u_3=(-4/9,-7/9,4/9)$ form an orthonormal basis of $\mathbb{R}^3$. Find the projection of any vector $x=(\xi_1,\xi_2,\xi_3) \in ...