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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Show there exists $\pi \in R_{++}^{nk}$ such that $A_{nk}(l)\pi \geq 0$ if and only if the determinant is non-negative

Let n,k be positive integers, with $n \geq 2$ and $k \geq 1$. Let l={${l_i}$} be an nk-dimensional positive real vector. Let $A_{n,k}(l)={(a_{i,j})}_{1 \leq i,j \leq nk}$ be the following $nk \times ...
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17 views

Sum of dot products of linearly independent vectors

Suppose I have $n$ column vectors of equal length $\{\vec{a}_i\}_{i \in \{1,...,n\}}$ which are linearly independent of one another. Suppose I have a further $n$ column vectors $\{\vec{b}_i\}_{i \in \{...
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18 views

Definitions of the operator norm of a matrix

I understand that the operator norm of a matrix is induced by the vector norm, and is given by $$ \|A\|_{\rm op} = \max_{\|x\| = 1}\|Ax\| $$ However, in a lecture, I see the operator norm being ...
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29 views

How to find unit vector with a positive first component?

Let $\mathbb R^4$ have the Euclidean inner product. Find a unit vector with a positive first component that is orthogonal to all three of the following vectors. u $= (1,-1,3,0)$, v $= (6,1,0,1)$, w $= ...
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$\partial(y^T*A*x)/\partial(x)$ Which one is True?

$\partial(y^T*A*x)/\partial(x)$ for the expression above I did the steps: $A*x = u \\ y^T*u = z \\ => \partial(z)/\partial(x) = \partial(z)/\partial(u)*\partial(u)/\partial(x) \\ =y^T*A $ but http:/...
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1answer
21 views

Proof about Rank of rows and columns being equal.

I was wondering if this reasoning could be considered a proof: Let $A$ be a matrix $(m \times n)$ with $m$ rows and $n$ columns. Now we consider that there are at least $k$ rows vectors that are ...
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1answer
18 views

Schur's lemma for finite-dimensional unitary representations

I am reading the book 'Representations of Linear Groups' by Rolf Berndt, and on page 19 they state the following theorem: 'Let $(\pi,\mathbb{C}^n)$ be a unitary matrix representation of a group $G$, i....
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Prove that if a linear combination of the rows equals the zero row AND a linear combination of the b's equals zero iff Ax=b has a solution.

Prove that if a linear combination of the rows equals the zero row AND a linear combination of the b's equals zero iff Ax=b has a solution. I can prove this in a roundabout way using $x^T (A^T y) = (...
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1answer
19 views

Eigen vector of the adjoint operator. Error in this approach?

Reisz Representation Theorem: If $V$ is finite-dimensional and $\phi$ is a linear functional on $V$. Then, there is a unique vector $v \in V~|~\phi(v) = \langle v,u\rangle~\forall~v \in V$ Definition: ...
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22 views

A matrix polynomial convergent to a matrices’ transpose

Is it true that for all $A \in \mbox{GL}(n)$ there exists $c_i$ such that the following holds? $$\sum^\infty_{i=0} c_i A^i = A^T$$ For symmetric matrices, it is simple. I suspect some rotation matrix ...
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37 views

Why a subspace is assumed instead of a vector space?

Why does the following theorem start with "Let $\{u_1, ..., u_p\}$ be an orthogonal basis for a subspace $W$ of $\mathbb{R}^n$" instead of "Let $\{u_1, ..., u_p\}$ be an orthogonal ...
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A question regarding matrix of inner product in the ordered basis from Hoffman Kunze

While studying Chapter - Inner Product Spaces from Hoffman Kunze, I have a question in section 8.1 . Adding it's image: How can I derive the condition that the matrix must satisfy the additional ...
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1answer
31 views

If $X$ is not full rank, are $X^TX$ or $X^TX + \lambda I_p$ invertible?

Suppose $X$ is a $n \times p$ matrix, where $rank(X) < p$. Since $X$ is not full rank, then it is not invertible. I'm trying to understand whether functions of $X$ are invertible: $X^TX$ $X^TX + \...
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18 views

A question in Corollary Section 7.1 of Hoffman Kunze Linear Algebra

I am self studying Chapter -7 of Linear Algebra from Hoffman Kunze and I have a question in 1st section in last corollary whose image I am adding . Image of Theorem 1: I have a question in 1st ...
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1answer
8 views

2 questions in Statement of Primary Decomposition Theorem and it's Corollary in Linear Algebra

I am self studying Linear Algebra from Hoffman Kunze and I have 2 questions in section 6.8 whose image I am adding-> Questions : (1) In the last paragraph how does one can deduce that $W_{i}'s $ ...
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12 views

A question based on invariant direct sums in Hoffman and Kunze

While self studying Linear Algebra from Hoffman and Kunze I have a question in section ( 6.7) of Invariant Direct Sums. Adding it's Image: [1]: https://i.stack.imgur.com/AnH3X.jpg Question is in ...
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14 views

Given inputs and output points from a linear transformation, how can we know when there are infinite linear mappings that satisfy the conditions?

Assume we have a transformation $T$ such that $T : \mathbb{R}^3 \mapsto \mathbb{R}^3$ where $$T\begin{pmatrix} 1\\ 1\\ -1 \end{pmatrix} = \begin{pmatrix} 2\\ 1\\ 0 \end{pmatrix},T\begin{pmatrix} 1\\ ...
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20 views

Does entanglement of a bipartite PPT state $\rho$ imply entanglement of $\rho + \rho^{\Gamma}$?

Consider an entangled bipartite quantum state $\rho \in \mathcal{M}_d(\mathbb{C}) \otimes \mathcal{M}_{d'}(\mathbb{C})$ which is positive under partial transposition, i.e., $\rho^\Gamma \geq 0$. As ...
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9 views

Calculate real matrix inverse of a complex matrix - extend

Given a Hermitian positive semidefinite matrix $A \in \mathbb{C}^{n \times n}$, and a matrix $D \in \mathbb{C}^{n \times 2n}$. Specially, matrix $D$ has the property that: $\{d_{ij}\neq{}0,\forall{}j=...
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1answer
40 views

Exponentiating similar Laplacians

Let $L_c$ be the $n\times n$ Laplacian matrix of the complete graph, and $L$ be the $n\times n$ Laplacian of any simple, connected graph possessing a vertex $k$ of maximum degree $d_k=n-1$. Clearly, ...
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1answer
35 views

Find whether this set is a subspace of $\mathbb{R}^3$

Given the following set $W$, determine if it is a subspace of $\mathbb{R}^3$ under the operations of addition and scalar multiplication defined on $\mathbb{R}^3$. $$W=\left\{(a_1,a_2,a_3)\in \mathbb{R}...
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1answer
25 views

Exercise about diffeomorphism induced on a subset of $\mathbb R^d$

I have to determine if the following maps $f: \mathbb R^d \to \mathbb R^d$ induce a diffeomorphism $Z \to Z$: $f:X \mapsto AX, \ A\in O(d)$ and $Z=S^{d-1}$; $f:X \mapsto AX, \ A\in O(d)$ and $Z=\...
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1answer
49 views

Calculate real matrix inverse of a complex matrix

Given a Hermitian positive semidefinite matrix $A \in \mathbb{C}^{n \times n}$. If $B=A^{-1},\ D=\text{Re}(B),\ C=D^{-1}$. where $D=\text{Re}(B)\Leftrightarrow{}d_{ij}=\text{Re}(b_{ij})$. Can I ...
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20 views

Bound on Gram Matrix composed by Lipschitz function

Assume we know that $f(x,y): \mathbb{R}^{a+b} \to \mathbb{R}^d$ is lispchitz w.r.t. $y$, i.e. $$\|f(x,y) - f(x,y')\| \leq L \|y-y'\|.$$ Is there a way to bound the product $f(x,y)\cdot f(x,y)^T \in\...
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1answer
31 views

'Shrinking' the Hilbert Space a $\mathrm{C}^*$-algebra is represented on

This is a follow on question to this. Suppose that there is a subspace $U\subset H$ a Hilbert space such that for all $f\in A\subset B(H)$ a $\mathrm{C}^*$-algebra, $f$ restricted to $U$ is scalar. ...
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1answer
69 views

Can we apply here the Cayley–Hamilton theorem?

We have the matrix \begin{equation*}A:=\begin{pmatrix}3 & 1 & 0 & -1& -1 \\ 0 & 2 & 0 & 0 & 0 \\ 1 & 0 & 2 & 0 & -1 \\ 0 & 0 & 0 & 2 & 0 ...
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2answers
55 views

Characteristic & minimal polynomial & geometric multiplicity

Let $\mathbb{K}$ be a field and $1\leq n\in \mathbb{N}$. Let $a\in M_n(\mathbb{K})$, such that $a_{ij}=0$ for all $i\leq j$. It holds that $a^n=0$. For $1\leq k\in \mathbb{N}$ we define \begin{...
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1answer
21 views

Showing $\operatorname{diag}(x)-xx'$ is positive definite on the tangent space of the unit simplex.

Let $x$ be in the unit simplex (i.e. $\sum_i x_i = 1, x_i \geq 0$ ). I want to show that $\operatorname{diag}(x) - xx'$ is positive definite on the tangent space of the simplex. That is, $z'[\...
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1answer
45 views

If $U$ and $T$ are one to one and onto then $UT$ is also - why is my proof not sound?

Let $V$, $W$, and $Z$ be vector spaces, and let $T: V \to W$ and $U: W \to Z$ be linear. Prove that if $U$ and $T$ are one to one and onto then $UT$ is also. I've seen other solutions to this problem ...
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17 views

Quadratic form can be represented as a convex combination of $\frac{n(n+1)}{2}+1$ ones

Question : In $\mathbb{R}^n$, consider an inner product $(\ ,\ )$. Here any linear map $L$ has the form $L(x)=(V,\ x)$ for some $V$ When $\|\ \|,\ \|\ \|^\ast$ are norms in dual relation, then we have ...
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1answer
14 views

Convergence of inner product of vector space of infinite sequence

I am trying to prove the convergence of the inner space in $\mathbb{C}^{\infty}$ Defined as $$ \langle a | b \rangle := \sum_{k=1}^{\infty}b_{k}^{*}a_{k} $$ Considering that every sequence satisfy $$||...
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1answer
33 views

What would be the arithmetic/algebraic rules for solving the problem $500=\frac{66}{\sqrt{1 - \frac{V^2}{(3 \times10^8)^2}}}$

Every direction I take to solve this problem leaves me with a negative on one side of the equation and $V^2$ on the other. Arithmetic/algebraic rules were the cause of my last question on this site, ...
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1answer
59 views

Why are isometries all and only $f:\mathbb{R}^n \rightarrow \mathbb{R}^n $ :$f(x)=Nx+p$,where $N$ ortogonal

A passage in my notes reads: Isometries in metric spaces coincide with the usual ones when considering mappings $f:\mathbb{R}^n \rightarrow \mathbb{R}^m $ with the respective Euclidean metrics. In ...
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1answer
73 views

Show $\log(\det(A))\le \operatorname{tr}(A)-n$

Suppose that $A$ is a real, symmetric, positive definite $n\times n$ matrix. Show that $$\log(\det(A))\le \operatorname{tr}(A)-n \quad \text{and} \quad \log(\det(I_n+A))\le \operatorname{tr}(A).$$ ...
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2answers
36 views

$v_i$ is an eigenvector for $T$ with eigenvalue $\lambda _i$ then it's eigenvector for $T^*$ with eigenvalue $ \bar{\lambda}_i$ given normal $T$

Given an inner product space $V$ and a normal operator $T$, prove that $\ker T=\ker TT^*$ The solution I found mentions that using the fact that $T$ is normal we know it is diagonalizable, so we have ...
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1answer
18 views

Comparing columns of exponentials of symmetric matrices

Let $A$ e $B$ be two symmetric matrices, with the additional property that the sum of all the entries of any column is zero. If the $k-$th column of $A$ equals the $k-$th column of $B$, what can be ...
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33 views

Can you square both sides of an equation containing matrices?

For example A=λI ⇒ A²=λ²I where A is square, and λ∈ℝ. Or more generally AB=CD ⇒ ABAB=CDCD. Assuming all the necessary matrix products are possible, what other conditions would need to be fulfilled for ...
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52 views

If $A \in M_n(R)$ be diagonizable with $tr(A^2)=0$ show A is the zero matrix

If $A \in M_n(R)$ be diagonalizable with $\operatorname{\mathbf{tr}}(A^2)=0$ show $A$ is the zero matrix If $\operatorname{tr}(A^2)=0$ then $\sum a_{i,i}^2+\sum2a_{i,j}a_{j,i}=0$ We also have $A^2=CDC^...
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33 views

Explain why if {v1, v2, …vn} is a set of linearly independent vectors in the vector space V and {u1, u2, …um} is a basis of V then m≥n

So i let n>m. {u1, u2...un} is a basis of V and {v1, v2,....vc} is linearly independent. I set the two equal and n>m. Since the set is linearly independent and dim(V), both sets are a basis of V....
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37 views

Show $\begin{bmatrix}d-\lambda\cr -c\end{bmatrix}$ is eigenvector of $\begin{bmatrix}a &b\cr c&d\end{bmatrix}$

Show $A=\begin{bmatrix}d-\lambda\cr -c\end{bmatrix}$ is eigenvector of $\begin{bmatrix}a &b\cr c&d\end{bmatrix}$ I first did $\operatorname{det}(A-\lambda I)=0$ and got $\lambda^2+(-a-d)\...
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33 views

Find all matrices that are invariant under base changes.

Today I was solving a problem about listing all pairs $(a,b)\in \mathbb{R}^2$ such tath there exist a unique symmetric $2\times 2$ matrix such that $det(A) = a$ and $trace(A)=b$, where $A$ is the ...
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4answers
92 views

Why do sometimes care for where vectors originate from and sometimes not? and exactly how many kinds of vectors are there?

When I did linear algebra in high-school, it wasn't of much importance where the vectors originated from and for me this is a really hard concept to grasp. It's like no matter where the two vectors ...
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1answer
39 views

is this a subspace [closed]

is the set of $f$ such that $f(0)=f(1)$ on $[0,1]$ a subspace? I think it is but I stuck on showing it.
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46 views

There is a relation between the eigenvectors of a matrix $A \in \text{SL}(2,\mathbb{R})$ and its transpose $A^T$?

I'm working with a function $F: \text{SL}(2,\mathbb{R}) \setminus \{ I, -I\} \to S^1$, where SL$(2, \mathbb{R})$ is the set of $2 \times 2$ real matrices with determinant $1$, $I$ is de identity ...
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0answers
14 views

Writing $\int_\Omega \nabla u^T M \nabla v$ in terms of $H^1$ inner product of $u$ with another function

Let $\Omega \subset \mathbb{R}^n$ be a smooth domain, $u,v \in H^1_0(\Omega)$ with the usual inner product and let $M=M(x)$ be a $n\times n$ matrix with entries $m_{ij}\colon \Omega \to \mathbb{R}$ ...
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37 views

What kind of Matrix Transformation is this?

Sorry if this is a very basic question, but I have been given a task to type a program in which I am supposed to convert square matrices into 4 different forms. I understand that one of them is a ...
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1answer
27 views

A question in a theorem of section 6.6 of Hoffman Kunze

I am reading Linear Algebra from Hoffman Kunze by myself and I have a question in a theorem of Section-" Invariant Direct Sums " : I am confused by how authors prove uniqueness of ...
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1answer
36 views

Eigenspace is a subspace of V - ψ is diagonalizable

Let $\mathbb{K}$ be a field and let $V$ a $\mathbb{K}$-vector space. Let $\phi, \psi:V\rightarrow V$ be linear operators, such that $\phi\circ\psi=\psi\circ\phi$. Show that: For $\lambda \in \text{...
2
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27 views

Deducing an equation after changing variables

Let $V$ be a graded vector space and for $j\geq 1$ let $m_j:V^{\otimes j}\to V$ be maps of degree $2-j$ such that for al $v\geq 0$ $$\sum_{v=j+q-1, j=r+1+t}(-1)^{rq+t}m_j(1^{\otimes r}\otimes m_q\...
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2answers
39 views

Show that $I$ is an ideal of $\mathbb{K}[x]$

Let $\mathbb{K}$ be a field, $x_1, x_2, x_3\in \mathbb{K}$ and $I:=\left \{f\in \mathbb{K}[x]\mid f(x_1)=f(x_2)=f(x_3)=0\right \}$. I want to show that $I$ is an ideal of $\mathbb{K}[x]$. So we take $...

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