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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

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Determinant of the matrix $I+A+A^2+ \cdots + A^{n-1}$.

Let $c_1, \ldots, c_n$ be real numbers and let $A = (a_{ij})_{n\times n}$ the matrix defined by $a_{ii+1} = c_i$ for each $i = 1,2, \ldots, n-1$, $a_{n1} = c_{n}$ and the other entries are zero. Show ...
Mauricio Urrego Vasquez's user avatar
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Defining $\mathcal{A}(S) = TS$ implies $\mathcal{A}$ and $T$ share the same eigenvalues. Need help with proof step.

The following exercise comes from Linear Algebra Done Right, 4th Edition, Sheldon Axler in Section 5A, exercise number 37. Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Define $\...
Paul Ash's user avatar
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Is this differential operator self-adjoint?

If $V$ is the space of functions in $C^{\infty}(\mathbb{R})$ such that $f(x)=f(x+2)$ for all $x\in \mathbb{R}.$ Equipping $V$ with the inner product of the space $C[-1,1]$, is the operator $D:f\mapsto ...
user926356's user avatar
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Find a matrix Q that has Q(Q^T)=I but does not have orthonormal columns

My textbook says that if a matrix Q is square and has orthonormal columns then Q(Q^T)=I, but it does not say the opposite (that if Q(Q^T)=I then Q has orthonormal columns). Is there an example of such ...
Bob Joe's user avatar
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2 votes
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Dual space isomorphism non-canonical choice example

In a lot of resources that I have read it is mentioned that the isomorphism between $V$ and $V^*$ is non-canonical, but I was never sure that I properly understood precisely what this means. I haven't ...
lightxbulb's user avatar
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Is there any solution to a system of inequalities $Ax>b$, where $x$ is a column vector and $A$ is a symmetric positive matrix? [closed]

So I need to solve the inequality $Ax>b$ where $x$ is a column and $A$ a symmetric positive matrix (in that case).
Amine Lahmouz's user avatar
1 vote
1 answer
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Independence of two functions

Assume that f is a continuous function over the interval $[a,b]$ and there exists $x \in [a,b]$ such that $f(x) \neq 0$. I want to show that $f(x)$ and $xf(x)$ are linearly independent on this ...
bruno's user avatar
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Existence of nested sequence of separable polynomials with roots in [0,1]

Question: Is therea sequence (ak), k∈N of non-zero real numbers such that, for all n ∈ N, the polynomial Pn is separable( split with simple roots) in [0, 1]? My thoughts : I guess we are invited to ...
Interloper's user avatar
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Maximum value of the element of matrix $f(M)$

$M \in M_n(\mathbb {Q}), rk(I-M) = tr (I-M), rk(M) = tr(M)$. What is the maximum value of the element of matrix $f(M)$, where $f = x^5 - x^4 + x^3 - x^2$? I am not sure, but I think that eigenvalues ...
Norman Bates's user avatar
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Inverse and Determinant of Matrix $Axx^TA+cA$

Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
温泽海's user avatar
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Let $u,v \in V$ such that $\phi(u) = 0 \implies \phi(v) = 0\quad \forall\,\phi \in V^{*}$. Show that $v = ku$ for some $k \in \mathbb{K}$.

Let $u,v \in V$ be vectors such that for every linear function $\phi \in V^{*} \implies \phi(u) = 0 \implies \phi(v) = 0$ holds. We need to show that $v = ku$ for some scalar $k \in \mathbb{K}$. In ...
João Vitor's user avatar
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What is $\bigoplus_{n\in \mathbb{N}} I^{n}$, is topology is Tychonoff subspace topology?

I want to know what is $\bigoplus_{n\in \mathbb{N}} I^{n}$ with $I=[0,1]$. Im not sure if is just: $$\bigoplus_{n\in \mathbb{N}} I^{n}=\{(x_{n})\in \prod_{n\in \mathbb{N}}I^{n}\mid \exists ~M\...
Yves Stanislas SH's user avatar
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$𝑉$ is finite-dimensional. $𝑈, 𝑊$ are subspaces of $𝑉$. $𝑉 = 𝑈 + 𝑊$. Prove that there exists basis of $𝑉$ consisting of vectors in $𝑈\cup𝑊$

I'm self-studying linear algebra and can't find solution to the problem to verify my answer. Suppose $𝑉$ is finite-dimensional and $𝑈, 𝑊$ are subspaces of $𝑉$ such that $𝑉 = 𝑈 + 𝑊$. Prove that ...
Keltar Helviett's user avatar
1 vote
1 answer
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About surjectivity of $V\cong (V^*)^*$

Let $V$ be a finite dimensional vector space. We know that $V\cong V^{**}$ naturally . For example, See $V$ is isomorphic to $V^{\ast\ast}$, the double dual space of $V$. My my question is that how ...
Mahtab's user avatar
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Laplace expansion for permanent of a matrix

How can we prove the Laplace expansion (the same that we use for calculating determinant of a matrix) for permanent of a matrix?
Jane Doe's user avatar
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What is the intuition to create an orthogonal design matrix in an iterative way (without Gram-Schmidt)?

Let $X_1, X_2, \dots, X_n$ be $n$ observations between $[-1,1]$ and $X_i \ne X_j$ if $i\ne j$. Let $\phi_0(x)=1$, $\phi_1(x) = > 2(x-a_1)\phi_0(x)$. When $r\ge 1$, $\phi_{r+1}(x)=2(x-a_{r+1})\...
Kaven Lin's user avatar
1 vote
1 answer
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$V$ Euclidean iff there exist linear isometry that $f(v)=w$ for $|v|=|w|$

From the book 'a course in metric geometry' exercise 1.2.24 : Let $V$ be a finite-dimensional normed space. prove that V is Euclidean iff for any tow vectors $v,w\in V$ such that $|v|=|w|$ there ...
hr1380's user avatar
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1 answer
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Derivation of the characteristic polynomial for homogenous difference equations

I'm struggling to find a resource that proves every solution to the homogenous difference equation $$ a_n = c_1a_{n-1} + c_2a_{n-2} + \dots + c_da_{n-d}. $$ is of the form $$ a_n = k_1n^{p_1}r^n + ...
viscous_cat's user avatar
-1 votes
1 answer
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Query regarding determinants - getting different results from different methods.

$ \begin{vmatrix} (1+a) & 1 & 1 & 1 \\ 1 &(1+b)& 1 & 1 \\ 1 & 1 &(1+c) & 1 \\ 1 &1 & 1 &(1+d) \\ \end{vmatrix} $ This determinant is solved ...
Qwerty's user avatar
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1 answer
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Eigenvalue of square of a matrix

If I am given a vector $w$, and a matrix $A$ such that $A^2w = 0$ and I know that $A$ is not nilpotent, is it correct to say that $Aw = 0$? Edit: Originally I had thought the above statement to be ...
Souroy's user avatar
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Determine the rotation necessary to bring a plane in contact with an ellipsoid

Given the ellipsoid $ (r - C)^T Q (r - C) = 1 \tag{1}$ And the plane $n^T (r - r_0) = 0 $. I want to determine the angle of rotation about an axis whose unit direction vector is $a$ and passes ...
i don't know what i am doing's user avatar
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Can we prove that ABA' is positive definite if B is positive definite and A is of full row rank

Suppose $A$ is a $r \times m$ matrix and $Rank(A) = r$, $B$ is a $n \times n$ symmetric matrix and $rank(B) = n$. Can we prove that $ABA'$ is positive definite? If it is not positive definiteness, ...
shani's user avatar
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Are there interesting LCH topological groups of matrices other than the classical groups?

The known classical compact groups of matrices are the groups: $U(N)$, $O(N)$, and $USp(2N)$. Are there other interesting (research-wise) LCH topological groups of matrices? References and a brief ...
a.e's user avatar
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3 votes
1 answer
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The rank of Sylow subgroup of special linear groups over finite fields

Let $p,\ell$ be two primes, and let $\mathbb{F}_{q}$ be a finite field of order $q=p^r$. We define the rank of a finite group $G$ to the smallest cardinality of a generating set for $G$. We denote by $...
stupid boy's user avatar
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1 answer
37 views

Negative eigenvalue in PSD sum of matrices

Suppose that $A+B$ is positive semidefinite. Then the sum of the two matrices has only nonnegative eigenvalues. I want to prove that, given $B$ is of rank $1$, $A$ has at most $1$ negative eigenvalue, ...
mtcicero's user avatar
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Characterizing the constancy of the row rank of a matrix function when the rank is everywhere less than the maximum

In the theory of public economics I have occasion to examine the behavior of the following "matrix function": \begin{equation*} Z(x) = \begin{pmatrix} 1 & 1 & \cdots & 1 \\ g_{...
HerbF1978's user avatar
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1 answer
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What algorithm can solve linear inequalities when dealing with matrices?

I naively assumed that the following inequality can be solved by taking a matrix inverse, wherein A is a matrix and, x and b are vectors - $Ax≥b⟺Ax=b+z,z≥0⟺x=A^{−1}b+A^{−1}z,z≥0$ $ z_i = (Ax)_i - b_i, ...
desert_ranger's user avatar
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0 answers
19 views

Extrema of the dynamical orbit of a matrix

Let's say I have a point $p$ in $\mathbb{R}^n$, and an $n \times n$ matrix $M$. Is there an analytic method— or fast-converging algorithm— to determine the extremal coordinates among all $p_k = M^k p$?...
trbabb's user avatar
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0 votes
1 answer
36 views

How to use Quadratic Form Reduction By using Gauss's Method on $q(x) = 7x_1x_2 + 8x_1x_3 + 4x_2x_3$

How to perform the Quadratic Form Reduction By using Gauss's Method on this: $q(x) = 7x_1x_2 + 8x_1x_3 + 4x_2x_3$? I understand the algorithm for when there's squares, but this one is particularly ...
Denis's user avatar
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1 answer
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Is it true that for square matrices such that $CBC^{-1} = PBP^{-1}$ then $C = \lambda P$ where $\lambda \in F$ for some field $F$?

I was solving a problem where for a non-zero linear operator $T$ and two matrices $A$ and $B$ such that these matrices are the matrices of $T$ relative to the ordered basis $\mathcal{B}$ and $\mathcal{...
Leaf's user avatar
  • 28
1 vote
2 answers
60 views

Is matrix idempotent? [duplicate]

Let't say we have a matrix A such that $rank(A)=tr(A)$ and $rank(I-A)=tr(I-A)$. I tried to prove $A^2=A$ given these properties but I failed. Yet I strongly suspect that this matrix A could be ...
user341's user avatar
  • 147
1 vote
0 answers
40 views

How to find the condition number of \begin{bmatrix} I & B\\ B^{T} & I \end{bmatrix}

I want to find the spectral condition number of this matrix, notice that this matrix is symmetric, hence the spectral condition number can be wriiten as $$\frac{\max_{1\leq i \leq n }|\lambda_{i}| }{\...
YuerCauchy's user avatar
1 vote
0 answers
24 views

Proving the Basis of a Scaled Vector Set J

If $ E = \left\{\overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}\right\} $ is a basis, prove that $ F = \left\{\alpha \overrightarrow{e_1}, \beta \overrightarrow{e_2}, \gamma \...
Gjhdby5 Vjfhu's user avatar
1 vote
1 answer
23 views

Proof of the Single Value Decomposition

I am working through the a proof of the single-value decomposition, from Strang's 'Introduction to Linear Algebra, 4th edition'. I have included the proof as shown in the book at the end of this post. ...
Joseph's user avatar
  • 373
-1 votes
1 answer
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Doubt related to Projection matrix

I came across Projection matrix recently.. If we consider matrix A with columns A1,A2,.....An forming basis for a subspace. Then the formula for projection matrix for projecting a vector onto column ...
newest newbie's user avatar
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0 answers
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Solving systems of linear algebraic equations in singular case

I want to solve this system: $A \in \mathbb{R}^{n \times n}, \quad Ax = b, \quad det(A) = 0$ Hence, there are either no solutions or there are infinitely many of them, then I want to find a pseudo-...
Mr Robot's user avatar
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0 answers
16 views

Finding parameter for positive semi-definiteness in the sum of two quadratic forms

here's a problem I've come across. It involves determining the values of a parameter $t$ for which a quadratic form $h$ is positive semi-definite. I've briefly discussed this with some friends, and ...
T0remyr's user avatar
10 votes
1 answer
41 views

Understanding Nash Equilibria in a Bimatrix Game

I am currently studying game theory and I came across a problem involving a bimatrix game. The bimatrix is given by: $$ (A, B) = \begin{pmatrix} (4, 2) & (0, 0) \\ (0, 0) & (1, 3) \end{...
鈴木悠真's user avatar
2 votes
0 answers
55 views

Find least squares system of linear equations

Find a real system of linear equations $Ax = b$ where A is 2 columns and 4 rows matrix and all elements of A are not zero, $b \in \mathbb{R}^4$, $z = (2\ 3)^T$ is the approximation of $Ax = b$ using ...
meerkat's user avatar
  • 395
1 vote
1 answer
39 views

Does the nonabelian group of order 21 embed into $GL_6(\mathbb{F}_3)$?

Does the general linear group $G=GL_6(\mathbb{F}_3)$ of six by six invertible matrices over the field of three elements contain the nonabelian group of order 21? It would suffice to find two matrices $...
primer's user avatar
  • 220
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0 answers
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Locus of center of hyperbola that is tangent to coordinate axes

Given the generic hyperbola $ \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $ which is centered at the origin, suppose we shift it and rotate it, such that one of its branches becomes tangent to the ...
i don't know what i am doing's user avatar
10 votes
2 answers
218 views

Necessary and sufficient condition for the strategy to be unique

Suppose $A$ is the $m \times n$ game matrix for a two-person zero sum game. Suppose row player uses the strategy $x\in P^m$, what is the condition for the strategy for the column player to be unique? ...
Provoke's user avatar
  • 193
6 votes
0 answers
104 views

Bounding $\|A^r - B^r\|$ for $r \geq 1$.

Let $A$, $B$ be positive semi-definite self-adjoint matrices. Is it true that, for $r \geq 1$, $r \in \mathbb{R}$, $$ \|A^r - B^r\| \leq rc^{r-1}\|A - B\| $$ where $\|\cdot\|$ denotes the operator ...
DimSum's user avatar
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0 answers
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The differential equation cannot be solved

Let $\Omega = (0, 1) \times (0, 1)$ (a "square"), $\vec{x} = (x_1, x_2)^T \in \Omega$, $\Gamma = \partial \Omega$, $$ \Omega_c \subseteq \Omega \left( \Omega_c = (0, 1) \times (0, \frac{1}{2}...
Dmitry's user avatar
  • 1,362
9 votes
1 answer
171 views

prisoner's dilemma bimatrix

I have a question about the following derivation Consider the prisoner's dilemma with the following bimatrix: $$ (A, B) = \begin{pmatrix} (-5, -5) & (-1, -10) \\ (-10, -1) & (-2, -2) \end{...
Provoke's user avatar
  • 193
-2 votes
0 answers
42 views

Check whether W is a subspace of V. [closed]

Consider the vector space $V = \Bbb R^3$. Consider the subset $W \subseteq V$ containing all vectors $u = (x, y, z)$ such that $x + y + z = 0$. Check whether $W$ is a subspace of $V$.
North-east India's user avatar
10 votes
2 answers
121 views

Bimatrix Game: Nash Equilibrium and Safety Levels

I am studying the following example but don't understand how the solution works: Consider the following bimatrix game: $$ (A, B) = \begin{pmatrix} 4 & 2 & 0 & 0 \\ 0 & 0 & 1 & ...
PowerPoint Trenton's user avatar
1 vote
0 answers
26 views

Stuck on proof of primary decomposition theorem for finitely generated torsion modules over PID

i'm stuck in the theorem 6.10 (Advance Linear Algebra - Steve Roman's Book), more specifically, in the following statement: If $M$ is a finitely generated torsion module over a Principal Ideal Domain ...
LeonhardEuler2006's user avatar
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0 answers
37 views

Every operator on a finite-dimensional vector space of dim ≥ 2 has invariant subspace of dim=2

I'm attempting to find a solution for this problem, and have written a proof and am unsure of a few steps, which I will mark. This is for self-study, so I appreciate the feedback. I'm also wondering ...
Cole's user avatar
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0 answers
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Calculation of the projection w from Linear Discriminant Analysis

In an assigment focused on Linear Discriminant Analysis(LDA) there is this theoritical exercise: A dataset has been derived from two classes $ \omega_A$ and $\omega_B$, the distributions of which are ...
Constantinos Pisimisis's user avatar

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