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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.

0
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1answer
19 views

Finding the inverse of a linear transformation

Let $A:P_3 \to P_3$ be linear operator such that $$Ap(x)=\int_0^1p(x+t)dt$$ where $p \in P_3$. Find $A(e)$ if $(e)=\{1,x,x^2,x^3\}$ and $A^{-1}(2x-x^3)$ I just started learning about linear operators ...
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4answers
66 views

Let $A \in \mathbb R^{5 \times 5}$. Prove that $A^{4} + I \neq O$

Let $A \in \mathbb R^{5 \times 5}$. Prove that $$A^{4} + I \neq O$$ I understand that I have to start from $A^{4}+I= 0$ and come up with a contradiction. Is it the Cayley Hamilton I have to use?
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votes
3answers
44 views

Intersection of subspaces of direct sum is only zero

If $U_1,U_2,\dots,U_m$ are subspaces of $V$. Then $U_1+\dots+U_m$ is a direct sum if and only if $U_1 \cap \dots \cap U_m={\vec{0}}$. Is this true or not?
1
vote
1answer
88 views

Prove that the determinant is greater than $1$

Let $A$ be an $n \times n$ matrix whose diagonal entries are strictly positive and off-diagonal entries are negative. The sum of the entries on each column is $1$. Prove that $\det(A) > 1$. I ...
0
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2answers
39 views

Spectral Radius $\leq$ min(1-norm, infinity norm)

How do I prove that the spectral radius of a matrix is less than or equal to the minimum of 1-norm and infinity norm of the matrix? i.e. $$\rho(A) \leq min(||A||_1, ||A||_{\infty})$$ I know the ...
4
votes
0answers
53 views

If zero is an eigenvalue are dimensions lost?

This is likely a silly question so sorry in advance. However, I am wondering if I am right in thinking that if zero is an eigenvalue, then some dimension must be lost. My understanding is that ...
3
votes
1answer
78 views

Why are eigenvectors important for Deep Learning applications?

I know it is quite of a trite question to ask about the importance of eigenvectors, but I do not understand how they can be relevant for Deep Learning and when we can use them. Any reference to the ...
2
votes
1answer
61 views

Gram-Schmidt Procedure from “Linear Algebra Done Right”

The following content is from "Linear Algebra Done Right" book by Sheldon Axler, 6.31. There was a part of the proof what I don't understand is that $\begin{align*} \left\langle e_j, e_k\right\...
2
votes
2answers
37 views

Finding a basis of the annihilator of a subspace

Let $V\subset \mathbb{R}^4$ be the subspace spanned by $e_1+e_2+e_3+e_4$ and $e_1+2e_2+3e_3+4e_4$. Find a basis of the subspace $V^{\circ}$ in the dual space $(\mathbb{R})^*$. My attempt: Let $a = ...
0
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1answer
17 views

Showing that a subset of a given space of matrices is a subspace

I've been given the following $V$: The set of $2 \times 2$ matrices with real entries $H := \{ A \in V : A \; \text{is symmetric} \}$ And I need to prove that $H$ is a subspace of $V$. So far, I ...
1
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1answer
66 views

Constructing quaternions - proof that square of each imaginary unit is -1

During construction of vector space of quaternions over real numbers I encountered a problem that I can't quite put my finger on. For the context: Hamilton a multiplication in plane that keeps ...
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0answers
14 views

Meaning of (viable) and (non viable) flow pattern?

I found all the unknown traffic flows for this network (check picture), but in the end, the question is asking to give (one viable flow pattern) and (one that is not viable) what does that mean? any ...
1
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3answers
76 views

Why is operator norm defined the way it is?

Is there an intuition for the infimum definition of ${\| A \|}_{\mathrm{op}}$ without using a different, equivalent definition? I am referring to the definition, given an operator $A: W \rightarrow V$...
1
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1answer
27 views

Real Inner Product Space, Hermitian Operator $T = S^{n}$ for n odd

Let V be a finite dimensional inner product space over $\mathbb{R}$, and $T: V\rightarrow V$ hermitian. Suppose n is an odd positive integer. Want to show: $\exists S:V\rightarrow V $ such that $T = ...
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0answers
26 views

Some doubts regarding the evaluation of the rank of a matrix

Here is a list of very novice questions that came across while studying: Suppose $A$ is an $m \times n$ matrix. Is the rank of $A\leq \min\{m,n\}$? Attempt: The "rank" of a matrix gives us the idea ...
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2answers
42 views

Is there any element of order 30 in $GL _{10}(\mathbb{C})$ with INTEGER coefficients?

I can easily find an element of order 30 in $GL _{10}(\mathbb{C})$ as $\begin{pmatrix} \cos\frac{2π}{30} & \sin\frac{π}{15} & 0 \\ \sin\frac{-π}{15} & \cos\frac{π}{15} & 0 \\ 0 & ...
1
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1answer
49 views

How to form Matrix pairs of $m$ friends and to find common friend from all possible pairs.

The below is a problem given in entrance exam. Problem: A golf club has $m$ members with serial numbers $1, 2 . . . , m$. If members with serial numbers $i$ and $j$ are friends, then $A(i, j) = A(j, ...
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2answers
102 views

Is a square zero matrix positive semidefinite?

Does the fact that a square zero matrix contains non-negative eigenvalues (zeros) make it proper to say it is positive semidefinite?
1
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0answers
27 views

Calculate a gradient

Given a constant $c\in\mathbb{R}^n$, and given $n+1$ constant vectors $u_0,...,u_n\in\mathbb{R}^n$ such that $u_1-u_0,...,u_n-u_0$ form a basis of $\mathbb{R}^n$ (ie they are affine-independant), let $...
1
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2answers
41 views

Proving the difference of two matrices is PSD

Claim: For $x\in \mathbb{R}^n$, we have $\operatorname{Diag}(x) - xx^T \succeq 0$ if and only if $x_i \geq 0 \ \forall i\in [n]$ and $\sum_{i} x_i \leq 1$. Where $\operatorname{Diag}$ denotes the ...
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0answers
21 views

Definition of a bilinear form?

Let $V$ and $A$ be abelian groups. Then, an $A$-valued bilinear form on $V$ is a $\mathbb{Z}$-module homomorphism $$\beta: V \otimes_{\mathbb{Z}} V \rightarrow A$$ What is this notation? What does $V ...
0
votes
1answer
30 views

Find an invertible matrix and a diagonal matrix

I have the following question. I know what the answer is supposed to be (I put it in mathlab). However, when I go through the individual steps I keep getting the wrong answer. Can someone tell me ...
1
vote
1answer
47 views

Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, …, 0, \operatorname{Tr(}A))$ respectively.

Let $A$ be a matrix $n \times n, n \geq 2 $. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar to a ...
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0answers
9 views

How to write this matrix multiplication in expression using sums

So I have to take a derivative of ANS with respect to $\lambda$ so (I think) have to write it in summation form. ANS = $y'UD(D^{2} + \lambda I_{p})^{-2}DU'y$, where y (1 x n), U (n x p), D (p x p). ...
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0answers
20 views

Finding iterative relationship between vectors, involving substitution of a product

Consider $A_N\in\mathbb{R}^{n\times n}$, $V=(\mathbf{v}_1,\mathbf{v}_2,\dots, \mathbf{v}_N)^T$, $\mathbf{v}_i, \mathbf{u}\in\mathbb{R}^n$, is it possible to express, $$\mathbf{v}_{N+1}=\frac{A_N^{-1}...
0
votes
1answer
21 views

Simultaneous diagonalization of matrices by using their product

If $A$ and $B$ are two simultaneously diagonalizable normal matrices, is it possible to find the common unitary matrix $U$ formed by their common eigenvectors by diagonalizing their product $AB$ since ...
1
vote
1answer
24 views

Why is the list $1,z,…,z^m$ linearly independent in $\mathcal{P}(\mathbb{F})$ (for $m \in \mathbb{N}$)?

This is an assertion made in Axlers Linear Algebra Done Right (2.18(d) in Chapter 2). $\mathcal{P}(\mathbb{F})$ of course is the set of all polynomials with coefficients in $\mathbb{F}$ and $\mathbb{...
2
votes
1answer
50 views

Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $.

Let $ A, X \in M_{nxn}(K) $. Let $ p_A(t) $ be a characteristic polynomial of matrix A. Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $....
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0answers
25 views

finding a matrix with respect to basis

$$B = \left\{1-x^{2},2x,1+2x+3x^{2} \right\} \; and\; B' = \begin{Bmatrix} \begin{bmatrix} 1\\-1 \end{bmatrix} \begin{bmatrix} 2\\0 \end{bmatrix} \end{Bmatrix} is \; [L]^{B'}_{B} = \begin{bmatrix} 2 &...
0
votes
1answer
31 views

Showing that $I - \alpha P$ has no zero eigenvalues when $P$ is a Stochastic matrix

I am trying to to show: $$ I - \alpha P$$ is non-singular and $\alpha \in [0,1)$. I know how to do it using nullspace arguments (see this) but I wanted to do it by showing the eigenvalues are NOT ...
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0answers
12 views

Jacobian of a skalar function with multi-dimentional vector input

I am trying to compute the Jacobian of $f : \mathbb{R}^{8} \rightarrow \mathbb{R}$, where: $f(\vec{x})= g(T(\vec{x}))= g(\vec{\mathbf{c}})=\Biggl| \|\mathbf{V}\|_{2}^{2} - \|\mathbf{A} \cdot c\|_{2}...
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0answers
52 views

Maximizing $x^T A^T B \, x$ over the unit Euclidean sphere

Is there an algorithm to solve the QCQP $$\begin{array}{ll} \text{maximize} & x^T A^T B \, x\\ \text{subject to} & \|x\|_2 = 1\end{array}$$ when $A^T B$ is not necessarily symmetric? When $A^...
0
votes
1answer
26 views

how to find a basis for ker(T)

I am struggling with an algebra problem here is what we got : $$ T\begin{bmatrix} a &b \\ c&d \end{bmatrix}= a+d\;\; \; \; and \; \; \;\; L\begin{bmatrix} a &b \\ c&d \end{bmatrix}=\...
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0answers
13 views

finding the left eigevector

I have the eigenvalue ${\cal{R}} _0$ of $P=\begin{pmatrix}0& \frac{\beta _h}{\mu +\gamma}&\frac{\beta _e}{\kappa (\mu +\gamma)}\\ 0& \frac{\beta _h\gamma}{(\mu +\gamma)(\mu +\alpha)} &\...
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votes
0answers
28 views

The Proximity Operator of a Function with Multiple Affine Mapping

Let $f(\mathbf{x}) = g(\mathbf{A}\mathbf{x})$, where $\mathbf{A} \in \mathbb{R}^{M \times N}$ is a linear transformation satisfying $\mathbf{A}\mathbf{A}^T = \mathbf{I}$. Then for any $\mathbf{x} \in \...
2
votes
2answers
54 views

The Projection of a Vector onto a Plane

I want to find the orthogonal projection of the vector $\vec y$ onto a plane. I have $\vec y = (1, -1, 2)$ and a plane that goes through the points \begin{align*}u_1 = (1, 0, 0) \\ u_2 = (1, 1, 1) \\ ...
0
votes
1answer
20 views

Proof of Batch Gradient Descent's cost function gradient vector

In the book Hands-On Machine Learning with Scikit-Learn & TensorFlow, the author only showed the formula for the Batch Gradient Descent method, such as: $ \dfrac{\partial}{\partial \theta_{j}} ...
1
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0answers
19 views

A question on vector norm error analysis

${x^*} \in {R^n}$ is the optimal solution of an optimization problem and leads to the minimum objective function ${\left\| {A{x^*} - b} \right\|^2}$, where $A \in {R^{m \times n}}$ and $b \in {R^m}$. ...
0
votes
1answer
33 views

A difficulty in understanding the n-dimensional second order derivative.

The example and its solution is given below: But I do not understand why in the calculation of $D^2 f(2,3)(u)^2$ the $u^2$ takes this form ....$ u_1^2 + u_{1}u_{2} + u_{2}^2$ from where the ...
1
vote
1answer
38 views

With these definitions how do I prove that this inner product is positive-definite?

Could someone please help me untangle the notation with the following? Let $V$ be a real vector space with inner product $\langle \cdot \,,\cdot \rangle : V \times V \rightarrow \mathbb{R}$. Let $W = ...
2
votes
2answers
22 views

Prove $(X \theta - \vec{y})^T (X \theta - \vec{y}) = \theta^T X^T X \theta - \theta^T X^T \vec{y} - \vec{y}^T X \theta + \vec{y}^T \vec{y}$

I'm studying Machine Learning Stanford's CS229 course and in the lecture note, page number 11, I'm not getting how does step 2 arrive from step 1 above? Prof. Andrew Ng says that it is the expansion ...
0
votes
1answer
28 views

Is the following function a linear transformation

I have the following exercise L : $\mathbf P_3 \to \mathbf P_2$ $ f \to 2f` + (f(3))t^2$ What I have tried: Let $f(s) = 3-s$ for an arbitrary $s$ and $a = 2 \in \mathbf R$. Hence, LHS: L($af$) =...
0
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1answer
28 views

projection onto affine image set

A={$y: y=Ax+b,A\in R^{n\times m}, x\in R^m$} I want to compute the projection onto A from a point z$\in R^n$,is there a close formulation to do that? Thanks.
0
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2answers
54 views

Shortest Distance between line and a point

L is the Line with parametric equation: x=2-t y=-4+t z=-2-t Find the shortest distance d from point =(3,1,1) to L and point Q on L that is closest to the point. ...
4
votes
1answer
34 views

In showing that if $V=X \oplus Y$, then $X \cap Y = \{0\}$, is writing $v=0+v$ and $v=v+0$ for $v \in X \cap Y$ sufficient?

$V=X \oplus Y$ if every vector in $V$ can be written uniquely as $v=x+y$ for $x \in X$ and $y \in Y$. Suppose $V=X \oplus Y$. We want to show $X \cap Y =\{0\}$. Suppose not. Let $v \in X \cap Y$. ...
0
votes
0answers
11 views

Prove that $y ∈ R(T)$ if and only of $φ_γ(y) = [y]_γ ∈ R(L_A)$, and that $x ∈N(T)$ if and only if $φ_β(x) = [x]_β ∈ N(L_A)$.

Let $V, W$ be finite dimensional vector spaces over $F$ with bases $β, γ$ respectively. For $T ∈ L(V,W)$, let $A = [T]_γ^β$. Prove that $y ∈ R(T)$ if and only of $φ_γ(y) = [y]_γ ∈ R(L_A)$, and that $x ...
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0answers
12 views

change of basis and linear transformation [closed]

enter image description here how to solve this question ? can I have detailed solution? thanks
0
votes
1answer
45 views

Dual space and Basis

I have a question. Given a finite dimensional vector space V, is the basis for the dual space of V also a basis for V?
2
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2answers
67 views

Show that $ S( a \times b) +(Sa) \times b \ + a \times (Sb) \ =0 $

Let $S$ be a $3 \times 3$ matrix which satisfies $S^T = S$ and $trace(S)=0$. Show for any $a,b \in \mathbb{R}^3$ that $$ S( a \times b) +(Sa) \times b \ + a \times (Sb) \ =0 $$ This can obviously ...
1
vote
0answers
44 views

Covariance matrix and projection

I have troubles understanding a geometrical meaning of a covariance matrix. Let's say we have a data set containing two points (-1,1), (-1,2) and write them in to the matrix $$D = \begin{bmatrix} -...