Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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Trying to understand Grobner basis

While studying Grobner basis, I realized that creating a basis from a given set of polynomials is not that hard, it is reduced to solving with Gauss Jordan a system of equations. What I don't ...
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Direct sum of two subspaces - how to show

I haven't found this exact question yet on this site: If $U$ and $W$ are subspaces of the inner product space $(\mathbb{R}, V, +, \langle .,. \rangle)$, and I have to show that $U \oplus W = V$, ...
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Distributing intersection over vector space addition.

Let $U, S, W$ be three subspaces of the vector space $V$. Prove or disprove the following a) $U ∩ (S+W) ⊆ (U ∩ S) + (U ∩ W)$ b) $(U ∩ S) + W ⊆ (U + W) ∩ (S + W)$ My Approach: Let $v ∈ U ∩ ...
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Solve System of equation using elimination?

\begin{align} I:&& ~~ x+\frac12y &= 6 \\[.5em] II:&& ~~ \frac32x + \frac{3}{2}y &= {17 \over 2} \end{align} when $x$ was multiplied by $(-3/2)$ in first equation the $x$ will ...
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Question regarding the similarity of an invertible matrix with its inverse .

Find the set $S$ of all possible $n×n$ invertible matrices $M$ such that $M$ is similar to $M^{-1}$ . My approach Actually, I was thinking about this problem when I came across a theorem stating ...
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Every matrix of the centralizer of the centralizer of a matrix is a polynomial in that matrix

Let $V=M(n,\mathbb C)$. For a subset $S \subseteq V$, let $C(S):=\{A \in V | AB=BA, \forall B \in S \}$ . How to prove that for every $A\in V$, we have $C(C (\{A\})) \subseteq \{ p(A) | p(t) \in \...
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$(C_{1}\cup C_{2})^{\perp}=C_{1}^{\perp} \cap C_{2}^{\perp}$, linear code $C_{1}, C_{2}$

Prove $(C_{1} + C_{2})^{\perp}=C_{1}^{\perp} \cap C_{2}^{\perp}$ for any linear code $C_{1}, C_{2}$ over $\mathbb{F}_{q}$ of the same length. we know $C^{\perp}= \{ x \in \mathbb{F}_{q}^{n}: <x,v&...
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For the purposes of DFT, is “the” primitive root of unity $w_n = e^{ 2\pi i / n }$ or $w_n = e^{-2\pi i / n }$?

I'm working on the section of Strang's Linear Algebra 4e that discusses discrete Fourier transforms. To make the exercises easier, I wrote myself a Python script that generates the $n$th Fourier ...
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Are the zero eigenvalues of a Laplacian matrix semi-simple?

It is known that the Laplacian matrix $\mathcal{L}$ for a directed weighted graph has at least one zero eigenvalue. If it has more than one zero eigenvalue, will there be non-trivial Jordan blocks ...
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Generalized eigenvalue problem for non-singular matrix

I'd like to solve a generalized eigenvalue problem of the form $$Ax=\lambda Bx$$ However, matrix $B$ is full-rank, Hermitian, positive semidefinite and satisfies $\mbox{Tr}(B)=a$. Matrix $A$ is ...
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How wide is the Birkhoff Polytope?

Define the width of a polytope $P \subset \mathbb R^d$ as the minimum length of the interval $\{v \cdot p:p \in P\}$ for $v$ in the unit sphere. In other words the width is the smallest number $W$ ...
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Positive-Definite Matrix Question

I want to prove that the matrix is positive definite using the fact that: If $A$ is symmetric and $\langle x, Ax \rangle$ > $0$ for a nonzero vector $x$ then $A$ is positive. So I have the ...
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58 views

Do $A$ and $A^T A$ share an eigenvector?

I have been learning about singular value decomposition from http://www.ams.org/publicoutreach/feature-column/fcarc-svd and they say that orthongoal vectors in the domain are mapped to orthogonal ...
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1answer
28 views

Trace of matrix $A^{\ast}A$

Given a $n \times n$ matrix $A$ with complex entries. And $A^{\ast}$ represents the conjugate transpose of $A$.Then If $\left | tr{\left ( A^{\ast}A \right )}\right | <n^2$, then $\left |a_{...
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How to compute (unipotent) radicals

My question follows some previous one, essentially this one. I want to understand, given an algebraic group $G$ (say linear), how to compute its radical and unipotent radical. The (unipotent) radical ...
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If $A$ is $n \times n$ non singular complex matrix and $B = (\bar A)' A$, where $(\bar A)'$ is the conjugate transpose of $A$ then…

If $A$ is $n \times n$ non singular complex matrix and $B = (\bar A)' A$, where $(\bar A)'$ is the conjugate transpose of $A$. If $x$ is an eigenvalue of $B$ then $x$ is real and positive. (True/false)...
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(A−λI)x=0 and x≠0 iff det(A−λI)=0: Why [[1,1],[1,1]][[2],[3]] = [[5],[5]] ≠ 0 when det([[1,1],[1,1]]) = 0?

As refered to Why non-trivial solution only if determinant is zero, I wonder why \begin{gather} \begin{bmatrix} 1 & 1 \\1 & 1 \end{bmatrix} \begin{bmatrix} 2 \\3 \end{bmatrix} = \begin{...
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Construction of tensor algebra: extending the multiplication from being defined on pure tensors

T.S.Blyth in his book Module Theory: An Approach to Linear Algebra defined multiplication on the tensor algebra $\bigotimes M = \bigoplus_{n \in \mathbb{N}}\bigotimes^n M$ by first defining it for ...
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Existence of a symmetric matrix $X$ such that $XBX = A$

let $A,B$ be two positive semi-definite $n\times n$-matrices such that $$\mathrm{Range}(B^{1/2}AB^{1/2})=\mathrm{Range}(B)$$ and $$\mathrm{Rank}(A)=\mathrm{Rank}(B)=n-1$$ so is there a real ...
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Is the space of maps which satisfy this vanishing condition finite-dimensional?

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Let $h:\mathbb{D}^n \to \mathbb{R}^{k}$ be smooth, and suppose that $h(x) \neq 0$ a.e. on $\mathbb{D}^n$. Set $$V_h=\...
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Understanding formula for projection of a vector onto a line

The formula for projection of a vector 'b' on line represented by vector 'a' is given as the following in Linear algebra and its applications by Gilbert Strang But why is that after finding the ...
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How to avoid “what goes up must come down” in motion equations

I am simulating a slot machine spinning and would like to start with an initial velocity and gradually apply a negative acceleration until the total displacement has reached the desired slot. The ...
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1answer
430 views

Dependence implies Wronskian being zero

From Wikipedia: If the functions $f_i$ are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes. Thus, the Wronskian ...
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Given similar $A,B$ matrices how to find a non-singular matrix $P$ such that $B=P^{-1}AP$.

Given 2 similar square matrices $A$ and $B$ of same order,how to get hold of a non-singular matrix $P$ such that $B=P^{-1}AP$.One way I know is to solve the system $PB=AP$ which often becomes hard to ...
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Is there a geometric interpretation about the euclidean distance between of 2 matrices?

The Euclidean distance between points p and q is the length of the line segment connecting them ($\overline{\mathbf{p}\mathbf{q}}$). $$\begin{aligned}d(\mathbf {p} ,\mathbf {q} )=d(\mathbf {q} ,\...
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1answer
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Showing the Matrix identity $A_{g\circ f,X,Z}=A_{g,Y,Z}\cdot A_{f,X,Y}$

We want to show the indentity in a specific way that I did not understand, the definitions and theorems which are stated in the proof will be listed. The book is Bosch Linear Algebra page 95, Th4: $...
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The Matrix of a reflection (around abitrary plane)

Let $\Upsilon :\mathbb{R}^3\rightarrow \mathbb{R}^3$ be a reflection across the plane: $\pi : -x + y + 2z = 0 $. Find the matrix of this linear transformation using the standard basis vectors and the ...
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220 views

Showing that the limit of non-eigenvector goes to infinity

Let $A$ be a $3$ by $3$ real matrix with the triple eigenvalue $1$. Also, further suppose its eigenspace corresponding to $1$ is only of dimension $1$. Thus, we can find a basis of $\mathbb{R}^3$, ...
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Least square solution based on the pseudoinverse solved efficiently with singular value decomposition

Hi apologies it's hard to type out the problem, I have a lecture slide on neural networks. It says the fitting error gives the matrix: N by M matrix of thi's multiplied by Mx1 weights minus Nx1 ...
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1answer
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Prove equivalence of two statements related the vector field $V$

Let $M$ and $N$ be linear subspaces of $V$. Prove that (a) if $y\in M$, $z\in N$ and $y+z=\theta$ then $y=z=\theta$ is equivalent to (b) if $y+z = y^{\prime}+z^{\prime}$, where $y,y^{\prime}\in M$ ...
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259 views

What properties must have two orthonormal matrix to commute?

What properties must have two orthonormal matrix to commute? I mean what properties must two orthonormal matrices $A$ and $B$ have for $AB =BA$?
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Characterizing linear maps with matrices

Bosch Linear Algebra page 92 We want to prove that the map $\psi:\operatorname{Hom}_K(V,W)\rightarrow K^{m\times n}$ with $f\mapsto > A_{f,X,Y}$ is an isomorphism. I have not understood why we ...
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Is there a basis which spans the real numbers?

Is there a finite set of real numbers $S=\{a_1, a_2, ..., a_n \}$ such that every real number can be written as a linear combination (with integer coefficients) of the elements of $S$? If no, is there ...
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Is it always possible to swap columns of a matrix by a left hand side multiplication?

I was thinking about swapping the columns of the matrix. It is well known that if you want to swap 2 columns of a matrix, you do a right hand side multiplication with a permutation matrix $T_{ij}$, ...
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A System of Equalities and Inequalities; Is There A Unique Solution?

Suppose $m_x,M_x,m_y,M_y\in\mathbb{R}$ are known and $x_l,x_h,y_l,y_h\in\mathbb{R}$ are unknown. Does the simultaneous system \begin{align*} x_l \hspace{1mm}+\hspace{1mm} x_h &\hspace{1mm}=\...
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Are the functions $\sin^2(x)$ and $\cos^2(x)$ linearly independent? $a\sin^2(x)+b\cos^2(x)=0$, such that the domain is the set $~\{0\}~$

Are the functions $\sin^2(x)$ and $\cos^2(x)$ linearly independent? $a\sin^2(x)+b\cos^2(x)=0$, such that the domain is the set $\{0\}$. (Only has the element zero). Because if you plug the value $0$ ...
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Singular Value Decomposition (SVD) - Odd step in proof

There are a lot of proofs about the singular value decomposition of a matrix $A \in \mathbb{R}^{m \times n}$. Now this is the start of the proof in my textbook: Take the symmetric matrix $A^T \cdot A$...
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Parameterizing rotation matrix

A general rotation matrix in terms of Euler angles is given by $$ \mathcal{R}=R_{\hat z}(\alpha)R_{\hat y}(\beta) R_{\hat z}(\gamma). $$ Working out the matrix multiplication we obtain the known ...
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Non-trivial $\operatorname{Hom}_R(M;N)$ [on hold]

In K. Conrad's handout on dual modules(https://kconrad.math.uconn.edu/blurbs/linmultialg/dualmod.pdf), there is the following theorem-exercise 2.11, which states: Let $R$ be an integral domain with ...
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Formula for rotating vectors

Let $u = (u_1, u_2)$ and $x=(x,y)$ a rotation of u by an angle θ. Then $\left\lVert u \right\rVert = \left\lVert x \right\rVert$ We know: $$cosθ = {{u \cdot x}\over \left\lVert u \right\rVert \cdot \...
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Why $A$ invertible $\iff \det A\neq 0$

Let $A$ a matrix $n\times n$ over $\mathbb R$. I'm trying to prove that $A$ is invertible $\iff\det A\neq 0$. If $A$ is invertible, there is $B$ s.t. $AB=I$, and thus $$\det(A)\det(B)=\det(AB)=\det(I)...
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Linear combinations of cash flows

We consider cash flow vectors over T time periods, with a positive entry meaning a payment received, and negative meaning a payment made. A (unit) single period loan, at time period t, is the T-vector ...
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1answer
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Embedding of $\mathrm{SU}(2)$ in $\mathrm{SU}(3)$

There is an embedding of $\mathfrak{su}(2)$ in $\mathfrak{su}(3)$ given by sending the standard basis $$ \begin{pmatrix}i/2&0\\0&-i/2\end{pmatrix},\quad \begin{pmatrix}0&-1/2\\1/2&0\...
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How to prove $(AB)^T=B^T A^T$

If $A$ is $m \times n$ and $B$ is $n \times p$ matrices, prove that $(AB)^T = B^T A^T$. Matrices' elements are $A = [a_{ij}], B = [b_{ij}]$. Let $C=AB=[c_{ij}]$, where $c_{ij} = \sum_{k=1}^n a_{ik}...
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Why we pick $0$ vector to check in linear indepenence?

I know that linear independence means each vector is not the linear combination of the others. But, I don't know why when we check whether a set of vectors are linearly independent, we only check for ...
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Prove that for a bilinear symmetric form $B$ there's a vector v$≠$0 that B(v,v)=0 iff $B$ and $-B$ are not positive

a bit of a messy question: Suppose there's a Bilinear symmetric form $B$ on vector space $V$ above $R$. how can I prove that there's a $v ∈ V$ that sustains $B(v,v)=0$ iff $B$ and $-B$ are not ...
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definite positive symmetric bilinear form

Let A be the matrix of a positive definite symmetric bilinear form. Prove ִִ$a_1,$$_1$$a_n,$$_n$≥$a$$_1,$$_n$$a_n,$$_1$ I dont really have any idea what to do here.
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There exists a smallest linear subspace that contains every $M\in\mathcal{M}$

This is Exercise 7 from Berberian (1992) Linear Algebra, p.23. Let $V$ be a vector space and let $\mathcal{M}$ be any set of linear subspaces of $V$. Let A be the union of the subspaces in $\mathcal{...
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4answers
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Matrixes of higher order like $M_{\aleph\times \aleph}$ [on hold]

I was wondering if thinking about matrixes of the form $M_{\aleph\times \aleph}$ and assigning properties to them like matrix multiplication makes sense and useful in some way. note: $\aleph$ is the ...