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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101 views

How can we solve this question without brute force

If $A\in GL_n(R)$, where $R$ is a commutative ring with identity, I would like to prove $$ M=\begin{pmatrix} A & 0 \\ 0 & A^{-1} \\ \end{pmatrix}\in ...
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32 views

Problem about real square matrix with rank 1

Given $A \in \mathbb{R}^{n \times n}$ and $\text{rank}(A) = 1$. By working only on real field, show that $A$ is diagonalizable if and only if $\text{tr}(A) \neq 0$. Here, $\text{tr}(A)$ is the sum of ...
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77 views

Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { ...
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49 views

Eigenvectors of distinct eigenvalues of symplectic matrix on $K^{2n}$ orthogonal?

I showed that for the standard hermitian form $\langle , \rangle _{I_n}$ on $\mathbb{C}^n$ the eigenvectors of distinct eigenvalues of a matrix associated to this hermitian form are orthogonal to each ...
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62 views

The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
3
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49 views

How to prove this identity?

Let $$ x_1=- \frac{b\, t}{\left(a + e\, t\right)}, \\ x_2=- \frac{c\, s\, \left(a + e\, t\right)}{\left(a\, b + a\, f\, s - b\, h\, s\, t + e\, f\, s\, t\right)}, \\ x_3=- \frac{q\, \left(a\, b + a\, ...
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48 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
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41 views

Let $A$ be an $n$ x $n$ matrix with complex entries such that Trace$(A)$ = $0$. Then $A$ is similar to a matrix with $0$ in the diagonal entries

Let $A$ be an $n$ x $n$ matrix with complex entries such that Trace$(A)$ = $0$. Then how to show that $A$ is similar to a matrix with $0$ in the diagonal entries?
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27 views

(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
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43 views

Row operations that change similarity class

Let $\mathbb{K}$ be a field and $A\in \mathcal{M}_{n\times n}(\mathbb{K})$ be a matrix. Which row operations on $A$ do not change its similarity class?
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71 views

Is there a name for the group of complex matrices with unimodular determinant?

Does the group $$ G = \left\{ A \in \mathbb{C}^{n \times n} : |\det(A)| = 1 \right\} $$ have a name? It obviously contains the unitary group $U(n)$ and the special linear group $SL(n,\mathbb C)$. ...
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52 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
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92 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
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76 views

An interesting system of equations

We have the following system with a and b, real numbers: $ax+y + z =4$ $x+2y+3z=6$ $3x-y-2z=b$ Show that $\forall a \in \mathbb{Z} $ there is a $b \in \mathbb{Z}$ such that the system admits a ...
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45 views

Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
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149 views

How can higher-dimensional projection maps be described mathematically?

New question: (resulting from discussions with Sabyasachi) I am wonder how can higher-dimensional projection maps, analogous to for example the Mercator, Miller, Behrmann projections, can be ...
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55 views

Condition for a particular matrix to be semi definite positive

Let $B$ a symmetric $N\times N$ real matrix whose diagonal elements are equal to one, that is to say $B_{i,i}=1$, $\forall i = 1, \dots N$ $B_{i,j} = B_{j,i}$, $\forall i,j = 1, \dots N$. $B_{i,j} ...
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32 views

Action of $\mathbb{F}_{p^2}^\times/\mathbb{F}_{p}^\times$ on $P^1(\mathbb{F}_p)$

Let $p$ be prime. Let $\alpha$ be a generator of the finite field $\mathbb{F}_{p^2}$. So, $\mathbb{F}_{p^2}=\mathbb{F}_p[\alpha]$. Multiplication by $\alpha$ is an $\mathbb{F}_p$-linear operator on ...
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228 views

What is the dual matrix (of a sample covariance matrix)?

Let $A$ be a matrix. I am most interested in the real, symmetric case, but for full understanding let's let $A$ be complex. What does it mean for $A^D$ to be the dual matrix of $A$? Can we interpret ...
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31 views

Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which ...
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57 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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46 views

What does the following matrix inequality mean?

Suppose $\{A_i\}_{i=1}^n$ is a collection of $m\times m$ matrices. I'm trying to understand the following criteria: There exists $\lambda\in[0,1)$ such that for all $x\in\mathbb{R}^m$ $$ \lambda ...
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106 views

Square root of positive definite nonsymmetric matrix

Let $N$ be a nilpotent matrix in $M_n({\mathbb R})$, such that $(I+N)^2$ is “positive definite” (but not necessarily symmetric) in the sense that $<X,(I+N)^2X>$ is positive for any nonzero ...
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43 views

Dynamical System of Matrices

Let $T_{nm}$ be the set of all possible binary rectangular matrices of dimension $n\times m$. The cardinality of $T_{nm}$ would be $2^{nm}$. Let f be a map from $T_{nm}$ to itself. Consider a ...
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46 views

Maximum and minimum value of $\|A\|_1$

Let $A$ be a $2\times 2$ real matrix. If $A$ is orthogonal, determine its maximum value and minimum value of $\|A\|_1$. My answer: Let $A=\begin{pmatrix} a \quad b\\ c \quad d \end{pmatrix}$. From ...
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51 views

How to prove that the determinant is the same no matter how you take it?

To find the determinant, pick a row and move along it creating minors and use the recursive definition of determinant. How do we know that the determinant will be the same no matter which row you ...
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65 views

Solving an equation with the form $Ax=b$

$$\begin{array}{l} \left( \begin{array}{l} \begin{array}{*{20}{c}} 1 & 2 & 3 & \cdots & n \\ \end{array} \\ \begin{array}{*{20}{c}} 2 & 3 & 4 & \cdots & ...
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41 views

On partition of matrix

Let $S \in \mathbb{R}^{n\times n}$ be positive definite matrix partitioned by $$S = \begin{pmatrix} S_{11} & S_{12} & S_{13}\\ S_{21} & S_{22} & S_{23}\\ S_{31} & S_{32} & ...
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70 views

Computing the decomposition of a representation of $S_n$

I have an explicitly defined representation of the symmetric group that I would like to decompose into irreducibles. How to do this most easily? The best approach I have so far is as follows: Find a ...
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67 views

Relationships between Reduced Row Echelon Form and the Fundamental Four Subspaces [inspired by Strang P143 3.2.34]

I'm trying to apprehend all the links between two matrices' RREFs and their $4$ fundamental subspaces. Does $RREF(A) = RREF(B) $ $1.1.$ $\implies null(A) = null(B)$? True because $null(A) = ...
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60 views

Find the rank of the matrix

Let $X,Y\in\mathbb R^n$ be two non zero (column) vectors. Let $Y^T$ denote the transpose of Y. Let A = $X Y^T$. What is the rank of $A$ and what is the necessary and sufficient condition for the ...
3
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148 views

Jordan canonical forms determined by a minimal polynomial

Find the Jordan canonical forms of all $9\times 9$ matrices over $\mathbb{C}$ with minimal polynomial $x^2(x-3)^3$. My method: each factor of the minimal polynomial corresponds to a type of Jordan ...
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101 views

Difference between Householder Reflections and Gram-Schmidt?

In numerical QR decomposition, when we calculate the orthonormal factor Q of a matrix, what is the difference in results if we use Householder Reflections to normalize the matrix or use Gram-Schmidt ...
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87 views

Non Existence of matrices $A,B\in M_n(\mathbb{R})$ such that $(I-(AB-BA))^n=0$

Question is to Prove: Non Existence of matrices $A,B\in M_n(\mathbb{R})$ such that $(I-(AB-BA))^n=0$. This question has already been asked already but then i am asking for clarification of another ...
3
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216 views

Why doesn't every square matrix have $n$ linearly independent eigenvectors? [Strang P310 6.1.26]

Curt Solution: First reason: The nullspace and column space can overlap, so $\mathbf{x}$ could be in both. Second reason: There may not be $r$ independent eigenvectors in the column space. Longer ...
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58 views

pullback of density

So I have been reading some differential geometry and they are talking about density's and they claim that a pull back of a density is a density but I only have a partial proof of why this is true. ...
3
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71 views

dot product between vector and matrix

In my book on fluid mechanics there is an expression $$ \boldsymbol{\nabla}\cdot \boldsymbol{\tau}_{ij} $$ where $\boldsymbol{\tau}_{ij}$ is a rank-2 tensor (=matrix). Given that ...
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45 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
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111 views

Question about orthogonal transformation / orthogonal matrices

I have a question about orthogonal transformations. If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product ...
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56 views

Deducing that polynomials span

Let us say that we are dealing with a countable family of polynomials with real coefficients in $n$ indeterminates that commute. Are there any known/common nice systematic ways to tell if their span ...
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48 views

Choosing an appropriate complete orthogonal basis

I have a function $f(x)$ which I want to represent as the sum over some complete orthogonal basis $\phi_i$ such that: $$ f(x) = \sum_{i} c_i \phi_i(x) $$ Where the $\phi_i$ are orthogonal with ...
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82 views

Basis for an infinite dimensional vector space.

Is there any good paper that focus on the topic of basis for infinite dimensional vector space that I can read/ study. I found some papers that mention about this topic online, but they are very brief ...
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145 views

Determinant proof

Let $A\in M_n(\mathbb C)$ and $\alpha \in \mathbb C$. If $B$ is the matrix obtained by multiplying a single row of $A$ by $\alpha$, then det$(B)=$ $\alpha$ det$(A)$. I'm trying to understand and use ...
3
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105 views

Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...
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50 views

Linearly (In)Dependency?

I really need some help with this proof.. Let V be a vector-space, $(A_i)_{i\in N}$ a sequence of linearly-independent subsets of V with properties: $ A_i \subseteq A_{i+1}$ (for all $in \in N$ ) ...
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1k views

Determinant of symmetric tridiagonal matrices

Given an $n\times n$ tridiagonal matrix $$A =\left(\begin{array}{ccccccc} ...
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69 views

Are solutions of $\frac{1}{2}(A^T+A)x=b$ and $Ax=b$ related?

I saw some statements about these 2 systems while I was reading something about linear algebra. So I am curious if the solutions of these 2 systems are related. If it is, how are they related? Thanks ...
3
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108 views

Eigenbasis and diagonal Matrix

Any transformation $T:V \rightarrow V$ can be cast into a diagonal matrix if there are $n$ distinct eigen-values for $T$, now it is said that $T$ becomes a diagonal matrix w.r.t. eigen-basis, does ...
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600 views

How can I solve system of linear equations over finite fields in WolframAlpha?

Is it possible to solve system of linear equations over finite fields using Wolfram Alpha? If yes, how can I do that? Let us take a system $x+y+z=0$, $2x+y+2z=0$, $x+3y+z=0$. If I want to solve this ...
3
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1k views

Possible eigenvalues of a projection matrix

What are the possible eigenvalues of a projection matrix? I know that a projection matrix is any square matrix $A$ such that $A^2 = A$. Thus, we obtain $$ A^2v = \lambda^2 v \\ Av = \lambda v $$ for ...