Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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
votes
1answer
23 views

A matrix equation involving eigendecomposition

Let $p<n$ and -$H\in\mathbb{R}^{n\times n}$ be symmetric with eigendecompoistion being $H=U\Lambda U^{\text{T}}$, -$A\in\mathbb{R}^{n\times p}$, -$D\in\mathbb{R}^{p\times p}$ be a diagonal ...
3
votes
2answers
51 views

How to show $\text{Tr}(M\log N)=\sum_{i,j}^n\lambda_i\log(\tilde{\lambda_j})(u_i^{\top}\tilde{u}_j)^2$?

The above question is the equation $(2.4)$ of the following paper: MATRIX EXPONENTIATED GRADIENT UPDATES. Let $M$ and $N$ be two $n \times n$ positive definite matrices where $M=U\Lambda U^{\top}$, $...
0
votes
2answers
31 views

Geometric Relationship between Two Vectors [duplicate]

Consider two column vectors such that $a = (1,2,3)^T$ and $b = (-3,3,-1)^T$. What is the geometric relationship between $a$ and $b$?
0
votes
1answer
38 views

Algebra Parameters

I got 4 equations, where are $x,y,z,u$ variables , and $a,b$ parameters. So the solution from the book is like this: $$x+(a+1)y-(a+1)z-au=1$$ $$ax+(a+1)y+az-2u=2$$ $$ax+(a+1)y-2z+au=b$$ $$(a-1)x+3(...
-1
votes
1answer
58 views

How to know if a matrix of a certain rank exists? [on hold]

Very confused by this question. There exists a $100\times 4$ matrix of rank $4$ such that $$A\cdot \begin{bmatrix} 1\\ 2\\ 3\\ 4 \end{bmatrix} =\vec0$$ True or false. Not sure ...
4
votes
0answers
31 views

Reference for Grassmann and Schubert varieties for Beginners .

I need some references to understand Grassmann and Schubert Variety as a beginner. I am looking for self-contained notes on these. Thanks.
0
votes
1answer
42 views

Find parameters that satisfy two conditions of an equation [duplicate]

There is the following equation. $$x^{2}+2(m-a)x+3am-2=0$$ a) Find $a$ such that the equation has real roots, $\forall m\in \mathbb{R}$ b) Find $m$ such that the equation has real roots, $\forall a\...
-1
votes
1answer
29 views

If $U_1,U_2\subseteq U, U$ linear independent set then $(<U_1> \cap <U_2>)= <(U_1\cap U_2)>$

If I take an element $v$ in $(<U_1> \cap <U_2>)$ why this element can be described as : $v=\sum_{i=1}^{k}\lambda_iz_i+\sum_{i=k+1}^{k+n}\lambda_{i}x_i=\sum_{i=1}^{k}\mu_iz_i+\sum_{i=k+1}^{...
0
votes
5answers
37 views

Proving whether the set of third degree polynomial is not a vector space?

In my book, it says the above set fails the first axiom. It says if I take two sets $p_1(x)=x^3-x^2$ and $q(x)= 1-x^2$. They are not closed under addition. I can understand why that's true by using a ...
1
vote
1answer
20 views

Let $S=\{ \mathcal{X_s} \;\;|\;\;\ s\in\mathbb{R} \}$. It is possible that $ Span(S) = \mathbb{R}^{\mathbb{R}}$

Let $S=\{ \mathcal{X_s} \;\;|\;\;\ s\in\mathbb{R} \}$. It is possible that $ Span(S)=\langle S\rangle = \mathbb{R}^{\mathbb{R}}=\{f:\mathbb{R} \rightarrow \mathbb{R} \; |\;\; f\;\;\ fuction \}$? I ...
1
vote
2answers
29 views

Proof that W is a subspace of V vectorial space

Let $ V = \mathbb{R}^3 $, considering $ W = \{ (x,y,z) \in V: x \leq y \leq z \} $. Is W a subspace of V? To proof that I have to demonstrate: 1) W is nonempty, that means $W \ne \emptyset$ 2) if $ ...
2
votes
1answer
20 views

Help to find a factorization of a matrix into a product and sum of matrices

if $\hspace{0.2cm}$$Z\in$ R$^{n\times n} $ $\hspace{0.2cm}$is the down-shift matrix with ones on the first subdiagonal and zeros elsewhere, and $L\in$ R$^{n\times n} $ $\hspace{0.2cm}$ is the lower ...
0
votes
0answers
11 views

Variety of submodules of a finitely presented module

Assume that $A$ is a graded $k$-algebra with $A_d=k$ in every degree $d$. Let $M$ be an $A$-module with finite free presentation $F_1\to F_0\to M\to 0$. I want to undertand the collection of finitely ...
1
vote
2answers
31 views

Finding values for which a linear transformation $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ satisfies $v_1, v_2 \in ker(L)$ and $L(v_3) = (1,1,1)$

I am working on an old exam question as preparation for my exam this coming week. The question states that $v_1 = (1, a, a^2), v_2 = (a^2, 1, a), v_3 = (a, a^2, 1)$. Part A asked to find the values ...
1
vote
0answers
30 views

Significance of this note about definition of determinant in Artin's Algebra

This question is from Artin Algebra, second edition. After defining determinant and proving many of its properties, author comments about possible alternative approach of defining determinant on P23: ...
0
votes
1answer
29 views

Can i prove that this matrix is PSD?

I have matrix $A \in \mathbb{R}^{N \times N}$, such that $A(i,j)=trace(B_iCB_j), \forall ij$. Matrices $B_i$ and C are PSD and symmetric with positive entries. Can I prove that $A$ is PSD too? In ...
0
votes
1answer
25 views

Linear Algebra, orthogonal columns and length

Suppose A is a 3x3 matrix whose columns are orthogonal and the length (two-norm) of each column equals 4. Then what is $A^T*A$? ...
1
vote
1answer
73 views

Let $A$ be complex matrices and $x$ be eigenvalue of $A\overline{A}$.

Let $A$ be non-singular $n\times n$ complex matrices and $x$ be negative eigenvalue of $A\overline{A}$. Show that algebraic multiplicity of $x$ is even number. I show that the $\det(A\overline{A})>...
0
votes
2answers
35 views

Find the line perpendicular to these two vectors

A line goes through point $A = (9, -7, 31)$ and the line is perpendicular to vectors $[7, 1, 2]$ and $[3, 0, 1]$. What is the equation of the line? The cross product of $[7, 1, 2]$ and $[3, 0, 1]$ ...
0
votes
1answer
39 views

Proving that $\sum_{i=1}^3\sum_{j=1}^3\sum_{k=1}^3\varepsilon_{ijk}a_{pi}a_{qj}a_{rk}=\det(A)\varepsilon_{pqr}$.

I want to prove the following identity: if $A$ is a $3\times 3$ matrix, then $$\sum_{i=1}^3\sum_{j=1}^3\sum_{k=1}^3\varepsilon_{ijk}a_{pi}a_{qj}a_{rk}=\det(A)\varepsilon_{pqr},$$ where $\varepsilon$ ...
0
votes
0answers
14 views

Respresentation of Vectors spanned by basis vectors in a column space. Is the representation correct

This question is related to coding theory. Consider a generator matrix $G$ of a systematic code. It is of the form $[I_k|P$. Any column in $P$ is a linear combination of some or all columns of$I_k$. ...
1
vote
0answers
20 views

Find $P_2$ and $Q_2$ where $P_2\cdot A \cdot Q_2=B$ knowing that $P\cdot A \cdot Q= I_r $

Lets assume we have three matrix $A$,$P$ and $Q$, where: $$P\cdot A \cdot Q= \left[ {\begin{array}{cc} I_r & 0\\ 0 & 0\\ \end{array} } \right] $$ Now find $P_2$ and $Q_2$ where: $$P_2\cdot A ...
0
votes
2answers
12 views

Diagonalizability in relation to squaring and transposition

True or False? Let A be a square matrix If $A$ is diagonalizable, then $A^2$ is diagonalizable. If $A$ is diagonalizable, then $A^t$ is diagonalizable. Re 1, my answer is that it is correct, but I ...
0
votes
1answer
42 views

$n \times n$ matrix associated with $F(X)=-X$

From S.L Linear Algebra: Find the matrix associated with the following linear map: $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ given by $F(X)=-X$ Considering that dimension of vector spaces ...
0
votes
3answers
48 views

Are all matrices almost diagonalizable?

Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form, $$ J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix}, $$ where $s$ is the eigenvalue (with ...
0
votes
1answer
16 views

Effect of squaring and the identity-matrix on eigenvalues and the characteristic polynomial

I am new to Linear Algebra, and would like some feedback regarding the following question: True or false? Let $A$ be a square matrix over $R$ If 3 is an eigenvalue of $A$, then 10 is an eigenvalue ...
0
votes
1answer
34 views

eingenvalues and adjoints

\begin{align}T_1 \,^* T_1 &= S_2 \,^* T_2\,^* S_1\,^* S_1 T_2 S_2\\ &=S_2 ^{-1}T_2 \,^*T_2 S_2 \end{align} This implies that $T_1\,^*T_1$ and $T_2\,^*T_2$ have the same eigenvalues (and ...
0
votes
1answer
20 views

With condition on linear functional f and g such that whenever $f(x)\geq 0$ then $g(x)\geq 0$ how to conclude about kernel of both?

V is finite dimensional vector space with $f,g$ are non zero linear functional with property that whenever $f(x)\geq 0$ then $g(x)\geq 0$ then show that 1) Ker $f\subset$ ker g 2)Ker $f= $...
0
votes
1answer
16 views

Question regarding orthogonality and linear independence

I am new to linear algebra and was a bit confused regarding the following… Any feedback would be really appreciated... True or false? If $K$ is a non-empty set of vectors in $R^n$, then $(K^\bot)^\...
1
vote
0answers
25 views

Find basis of $ker f*$ and $im f*$

I have such information: $f: \mathbb R^{4} \rightarrow \mathbb R^{3}$, matrix $M(f)^{B}_{A}$ and I know vectors in basis $A$ and $B$.Do I think that I have to set $M(f*)^{A*}_{B*}=M(f)^{B}_{A}$and ...
0
votes
1answer
14 views

Question regarding the algebra of norms

I am new to linear algebra, as am having some doubts regarding the following question: True or False $u,v \in R^n$ $\left\lVert u\right\rVert=\left\lVert v\right\rVert$ if and only if u+v and u-v ...
1
vote
1answer
10 views

Two statements regarding orthogonal unit vectors and orthogonal complements respectively

I am new to linear algebra, and I was confused regarding the following question. I would really appreciate it, if anybody could give some feedback... True or False? $\left(\frac{1}{\sqrt14},\frac{-2}...
1
vote
2answers
24 views

Markov matrices with no ones on off-diagonals don't have -1 as an eigen value

Every time I saw a Markov matrix with an Eigen value of -1, it had some 1's on its off-diagonals. The most obvious example is a simple permutation matrix: $$M = \left(\begin{array}{ccc}0&1\\1&...
0
votes
1answer
20 views

Basis functions and positive definiteness

Suppose I have a set of functions $g_i(x)$ which form a basis on some interval $[a,b]$. My question is whether the matrix $$ A_{ij} = \int_a^b g_i(x) g_j(x) dx $$ is positive definite? My approach so ...
0
votes
1answer
34 views

A necessary condition for linear dependence?

Given the vector space $\mathbb{C}^n$ over $\mathbb{C}$. A sufficient condition for the set of vectors $$ \{ (1, x_1, x_1^2, \dots, x_1^{n-1}), (1, x_2, \dots, x_2^{n-1}), \dots, (1, x_n, \dots x_n^{...
0
votes
1answer
21 views

Matrices with same row reduced form - Show there's a sequence of row operations

"Let us assume that Q and W are z x y matrices such that they have the same row reduced form. Prove that there exists a sequence of row operations that takes us from Q to W." How would one prove this ...
6
votes
1answer
94 views

what does it mean by determinant of Jacobian matrix = 0?

I have an example: $$ u={x+y\over 1-xy} $$ $$ v = \tan^{-1}(x)+\tan^{-1}(y) $$ So by calculating the determinant of the Jacobian matrix I get zero. Does it mean there is no functional relationship ...
2
votes
1answer
12 views

Direct sum and intersection of more than two subspaces

In "Linear Algebra Done Right" Axler points out that if you have more then two subspaces, it is not enough to test that each pair intersects only at 0, in order to say that they have direct sum. As an ...
1
vote
1answer
20 views

Find a determinant of a matrix $M_{(n)x(n)}$ depending on the $x,y,n \in \mathbb R$

I had a matrix which I I transformed to zero as many matrix elements as possible and in this time I have: $$y^{n} \cdot det \begin{vmatrix} \frac{x}{y}+1 & 1 & ... & & & 1\\ \...
1
vote
1answer
60 views

Prove that there are no natural numbers $x$ whose digits are $0$ or $2$, such that $x$ is a perfect square.

Prove that there are no natural numbers $x$ whose digits are $0$ or $2$, such that $x$ is a perfect square. I need some help here. I thought starting with $x = n * 10^k$ where $10^k$ represents the ...
6
votes
1answer
46 views

Representation of negative Quantum entropy in terms of eigenvalues, i.e., $\text{Tr}(M\log M -M)=\sum_{i=1}^{n}(\lambda_i\log(\lambda_i)-\lambda_i)$?

Negative Quantum entropy or Negative Von Nuemann entropy is defined as $f(M)=\text{Tr}(M\log M -M)$. Where $M$ is a positive definite matrix in $\mathbb{S}_+^n$, $\log$ is natural matrix logarithm ...
1
vote
1answer
34 views

Determinant of a large symmetric block matrix

Consider a given matrix $Q \in \text{Mat}_N(\mathbb{R})$, which is invertible, and $n \geq 1$. I am looking for the determinant of the symmetric block matrix $I_n(Q)$ of total size $nN \times nN$: $$...
0
votes
0answers
18 views

Binary Polymatroid Optimization Problem

Let $\mathcal{N}$ denote the finite set $\{1, 2, \ldots, n\}$, and let $\mathcal{S}_j$ denote the set $\{1, 2, \ldots, j\}$; let $f\colon \mathcal{N} \to \mathbb{N}$ be nondecreasing, submodular and ...
2
votes
1answer
27 views

Which one is most cost expensive to solve a linear equation? LU or inverse?

Which one is the most expensive way to solve for linear equation? LU-decomposition $$A = LU$$ Or finding the inverse $$A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A)$$ If I have to choose, I ...
-1
votes
2answers
71 views

If $K= \mathbb Z /2\mathbb Z$ what is meant by $K^2$

I realize that $K= \mathbb Z /2\mathbb Z$ is simply the set of equivalence classes. But I recently came across $K^{2}$ given $K= \mathbb Z /2\mathbb Z$ It is then stated that $K^{2}=\{\{0,0\},\{1,0\},...
-1
votes
2answers
19 views

Linear Equations with three variables [on hold]

Suppose $p(x)=ax^2+bx+c$ is a polynomial where $a, b$, and $c$ are fixed but unknown constants. Setup and solve a $3 \times 3$ system of linear equations to find $a,b$, and $c$ if we know that $p(1)=4,...
0
votes
1answer
22 views

Understanding the defintion of dual operators

I'm reading a book about Functional Analysis, and now I've reached to the part about dual operators. I'm having some difficulties understanding the following definition - Why $A^*$ is $Y^*\...
2
votes
0answers
32 views

Why are $(1,0)$ and $(0,1)$ tensors antisymmetric?

The book I'm reading (Nadir Jeevanjee (auth.)-An Introduction to Tensors and Group Theory for Physicists-Birkhäuser Basel (2015)) defined an antisymmetric tensor of type $(r,0)$ or $(0,r)$ as "one ...
0
votes
1answer
31 views

Cauchy-Schwarz inequality for points on unit sphere?

I have the following problem (I added a photo of the problem): For a set of points $x \in \pi_k$ with $x \in R^d$ and on unit sphere we compute: $m_j = \frac{1}{n_j}\sum_{x \in \pi_k} x $ and ...
1
vote
0answers
18 views

How to verify the rank-nullity theorem for the linear application $\frac{d}{dx}\in \operatorname{End}(\mathbb{K}[x]_2)$?

I know that the rank-nullity theorem states that let $T: V\rightarrow W$ be a linear transformation, then: $$ \operatorname{Rank}(T)+ \operatorname{Nullity}(T)=\dim(V)$$ But since $\frac{d}{dx}\in ...