Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

learn more… | top users | synonyms

0
votes
1answer
12 views

Linear algebra eigenvalue proof

I asked this question before but after thinking about it I was trying to think of another way. Proof multiplied complex matrix has non negative eigenvalues I need to show given that A is a matrix $\in ...
0
votes
3answers
27 views

Formulation of the matrix determinant

How was the idea (and the equation) for the determinant of a square matrix formulated, and why does it work? All I've learned is that the determinant of a matrix is 0 when some row is a linear ...
0
votes
1answer
14 views

Proof multiplied complex matrix has non negative eigenvalues

I need help to show given that $\mathbf A$ is a complex nxn matrix that $\mathbf {AA}$* is a Hermitian matrix and the eigenvalues > 0 Where * is taking the transpose of the complex conjugate of the ...
-1
votes
0answers
25 views

Prune a linearly independent set? What is the element of Span(Z)? What is the

Consider the following subset of $P_3(\mathbb{R})$ (real polynomial functions of degree at most 3). $$ Z = \{f_1, f_2, f_3, f_4, f_5\} $$ where $f_1(x) = 1-2x+2x^2-x3$, $f_2(x) = 1-x+x^2+x^3$, ...
0
votes
0answers
10 views

Solution space for a set??

Make a system with 3 eqns and 3 variables of which the solution space is spanned by V =[ 1 3 0], [1 0 -1]? Do I do the cross product of these two? Can someone show me how to do this? Also Can you ...
0
votes
0answers
10 views

dimension of vector space $\frac{\langle e_{ab_1\ldots b_p}\rangle}{\langle \sum_{1\leq i\leq p}e_{ab_1\ldots \widehat{b_i}\ldots b_pc}\rangle}$

Let $p$ be a prime and $n\!\in\!\mathbb{N}$. What is the dimension of the $\mathbb{Z}_p$-module $$V_{p,n}=\frac{\langle e_{ab_1\ldots b_p};\: 1\leq a<b_1<\ldots<b_p\leq n\rangle}{\langle ...
0
votes
1answer
14 views

This is only a subspace if $b=0$ - Axler - LADR p13

I have written here in Axler - Linear Algebra Done Right, page $13$. If $b\in \mathbb{F}$, then $\{(x_1,x_2,x_3,x_4)\in \mathbb{F}^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb{F}^4$ if and only ...
3
votes
0answers
25 views

Find the determinants of the given matrices

Consider scalars $a,b,c,d,e,f$ such that $\det\left( \begin{array}{ccc} a & 1 & d\\ b & 1 & e\\ c & 1 & f\\ \end{array} \right) = 7$ and $\det\left( \begin{array}{ccc} a & ...
0
votes
0answers
11 views

Lovasz number and bipartite complement

Let $G=(V,E)$ be a graph on $n$ vertices. An ordered set of n unit vectors $U=(ui|i∈V)⊂R^N$ is called an orthonormal representation of G in $R^N$, if $u_i$ and $u_j$ are orthogonal whenever vertices i ...
0
votes
1answer
15 views

Proving Equality of Hermitian matrix

I need to show that the equality $\ A \mathbf u \cdot \mathbf v = \mathbf u \cdot A\mathbf v \ $ holds true for all $\mathbf u, \mathbf v \in \mathbb{C^n}$ if and only if the matrix A is Hermitian ...
3
votes
5answers
108 views

Is the determinant of this matrix positive or negative?

$\left( \begin{array}{ccc} 1 & 1000 & 2 & 3 &4\\ 5 & 6 &7&1000 &8\\ 1000&9&8&7&6\\ 5 & 4&3&2&1000\\ 1&2&1000&3&4\\ ...
2
votes
1answer
23 views

Every skew-symmetric matrix has a non-negative determinant

I'm breaking this up into the even case and odd case (if $A$ is an $n\times n$ skew-symmetric matrix). So when $n$ is odd, we have: $\det(A)=\det(A^T)=\det(-A)=(-1)^n\det(A)\Rightarrow ...
1
vote
1answer
13 views

Basis and dimension of the subspace of solutions to $A\mathbf{x}=\mathbf{0}$

Consider $$ A =\left( \begin{matrix} 1 & -1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ \end{matrix} \right) $$ and find a basis and the dimension of $S(A,0)$, where $S(A,0)$ is the ...
0
votes
0answers
15 views

Link between the projection onto a subspace and projection onto hyperplane

The projection onto a hyperplane $H=\{x\in \mathbb{R}^n|\langle a,x\rangle=b\}$ is defined to be $P_{H}(x)=x-\frac{\langle a,x\rangle-b}{||a||^2}a,$ and characterized by $\langle c-p,x-p\rangle\leq0.$ ...
1
vote
0answers
9 views

Double circle representation using ideal rubber bands

Let $V$ be a 3-connected planar graph, $V^*$ be its dual graph. Let $(C_i : i \in V)$ and $(D_p : p \in V^*)$ be a collection of circles so that if any edge $\{i,j\}$ borders any faces $a$ and $b$, ...
1
vote
2answers
34 views

Proof that every subspace is a vector space

I was unable to find a simple proof that a subspace is a vector space. I know that a subspace $S$ is a subset of a vector space, such that: $$\vec 0 \in S\\\vec a + \vec b \in S\\\alpha\vec a \in S$$ ...
0
votes
1answer
53 views

Equation. Proof that there is no solution.

Prove that $\nexists x,y \in \mathbb{N} $ such that: $$\begin{cases} x -y - 3k -1 = 0 \\ x-y -4l -2 =0 \\ x+y - 3f - 2 =0\\ x + y - 5m - 2 = 0 \\ x,y \in \mathbb{N} \end{cases} $$ I'm asking form any ...
0
votes
0answers
15 views

Prove that $L(V,W)$ is a vector space over $K$

Let $V$ and $W$ are $k$-vector spaces and let $L(V,W)$ be the set of linear maps $V\to W$. I have to prove that $L(V,W)$ is a vector space over $K$. I have already done a step by step proof ...
0
votes
0answers
21 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
1
vote
1answer
19 views

Prove a certain matrix is positive semidefinte.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi$ to be a diagonal matrix with a principal left eigenvector of $P$ on the main diagonal and zeros ...
0
votes
1answer
19 views

Show matrix is element in eigenspace

Let $A$ be an $n\times n$ matrix such that $A^2=A$. a) Let $E_{1}(A)=\{x \in \mathbb{R^n} | Ax=x \}$: let $E_{0}(A)=\{{ x \in \mathbb{R^n} | Ax=0\}}$. Let $x$ be any vector in $\mathbb{R^n}$. Show ...
4
votes
5answers
117 views

Rank of a matrix $A^2$ without calculating the square

I have a matrix $A=\begin{bmatrix} 2 & 0 & 4\\ 1 & -1 & 3\\ 2 & 1 & 3 \end{bmatrix} $ with rank 2. How do I prove that the matrix $A^2$ has also rank 2 without actually ...
0
votes
0answers
4 views

Find coordinates of point that satisfy given conditions

I have A(1,2,3) , B(-1,0,1), C(1,-1,1) which are points in $\mathbb{R}^3$. I'm trying to find another point H such that AH${\parallel}$AC and BH${\perp}$AC. I set H = ($h_1$, $h_2$, $h_3$), and took ...
1
vote
1answer
16 views

Spectral decomposition - generalized eigenspaces

Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i ...
1
vote
1answer
22 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
1
vote
0answers
12 views

Eigendecomposition Parameterization of Real Matrix

Given a set of distinct non-real eigenvalues $\lambda_1, \dots, \lambda_N$, so that $\lambda_{2n} = \overline{\lambda_{2n+1}}$. Accordingly given a set of non-real orthonormal eigenvectors $v_1, ...
2
votes
0answers
34 views

Orthogonal basis of $R^n$

I have to prove that if orthogonal basis of $\mathbb R^n$ containing only vectors which coordinates are $1$ or $-1$ exisists then $n \leqslant 2$ or $n$ is divisible by 4. It's obvious that n have to ...
0
votes
1answer
17 views

Help solving system of linear equations.

In the process of running through an algorithm, I have derived the following systems of equations: i) $1/3 + 1/3x_1 + 1/3 x_6 = x_5$ ii) $1/2 + 1/4 x_6 = x_1$ iii) $1/2 + 1/2 x_5 = x_6$ I've tried ...
1
vote
2answers
49 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
1
vote
1answer
21 views

Which of the following are true?

I need to find which of the following are true? $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$ My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim ...
2
votes
1answer
21 views

Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$.

Let $H=L^2(a,b)$ with $a<b$. Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$. Verify also that $$\{(b_1-a_1)^{-\frac{1}{2}} \cdot \dots ...
2
votes
1answer
34 views

understanding the matrix transpose

Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear ...
-2
votes
4answers
18 views

Orthogonal nonzero vectors and linear independence. [on hold]

Show that if $\mathbf{u}$ and $\mathbf{v}$ are orthogonal nonzero vectors, then they are linearly independent.
0
votes
3answers
47 views

Question on matrix exponential

Let $A$ be a real matrix with real eigenvalues $\lambda_k$ and complex eigenvalues $\alpha_ k \pm i\omega_ k$ , all of which are simple. I'm trying to show that every element of the matrix $e^ {tA}$ ...
5
votes
4answers
84 views

How do I show that the derivative of the path $\det(I + tA)$ at $t = 0$ is the trace of $A$?

Here, I'm taking $A$ to be a linear operator on $\mathbb R^n$ for $n>1$. Can you please tell me how to solve such a problem?
-1
votes
1answer
30 views

Show that $W$ is a subspace of $\mathbb{R}^n$ [on hold]

Suppose $\mathbf{z}$ is a vector in $\mathbb{R}^n$ and $W = \{\mathbf{u}\in\mathbb{R}^n:\mathbf{u}\cdot\mathbf{z}=0\}$. Show that $W$ is a subspace of $\mathbb{R}^n$.
0
votes
0answers
8 views

One-to-one correspondence between mean value and parameters

I am currently taking a course in statistics, and in this course we are considering linear models $\mu = X\beta$ where $\mu \in L$ and $L = col(X)$ is a linear subspace of $\mathbb{R}^n$, $X$ is the ...
2
votes
1answer
27 views

Reduce matrix to Smith Normal form.

I've been given the finitely generated abelian group: $$\langle x_1, x_2 \mid 6x_1-6x_2, -6x_1-12x_2, 4x_1-8x_2\rangle$$ and written the corresponding matrix: $$A=\begin{pmatrix} 6 & -6 \\ -6 ...
1
vote
1answer
23 views

Maple - Substitute into an expression involving derivative

I'm working with some matrices including derivatives like d/dt(x(t)). I need to replace this whole expression {d/dt(x(t))} with something like xdot. I have tried to use "subs", but it seems to refuse ...
0
votes
0answers
10 views

Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
4
votes
2answers
36 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
2
votes
2answers
18 views

Relation between the determinant of a linear mapping and norm of a multiplied vector

Let $S,T:\mathbb{R}^n \to \mathbb{R}^n$ be linear mappings with $\|Sv\|\le \|Tv\|$ for all $v\in\mathbb{R}^n$. Is it generally true that $|\det(S)|\le |\det(T)|$?
1
vote
1answer
8 views

P - bounded polyhedron, L - linear map. Show that L(P) is a bounded polyhedron

Let $P = \{x\in \mathbb{R}^n \ | \ Ax\leq b\}$ be a bounded polyhedron. Let $L:\mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear map. Show that $L(P):=\{L(x)\ | \ x\in P\}$ is a bounded polyhedron. ...
1
vote
0answers
36 views

$(v_1,v_2,v_3)$ is positively oriented iff $\det(A) > 0$

Suppose $(v_1,v_2,v_3)$ are three linearly independent vectors in $\mathbb{R}^3$. Suppose $A = \left( \begin{array}{ccc} | & | & | \\ v_1 & v_2 & v_3 \\ | & | & | ...
0
votes
0answers
10 views

How to show that the rotation around $m_0$ about an angle $φ$ is an affine map? [on hold]

Let $E^2$ be the two-dimensional Euclidean space (Euclidean plane), and let $m_0 = (1, 0)$. Show that the rotation around $m_0$ about an angle $φ$ is an affine map. Give a formula for this map with ...
1
vote
0answers
28 views

How are $EA$ and $AE^T$ related, where $E$ is an elementary matrix?

If $E_{n\times n}$ is an elementary matrix, and $A_{n\times n}$ is any matrix, how are $EA$ and $AE^T$ related? I understand that $EA$ is a row operation on $A$ and $AE^T$ is the same operation on ...
-1
votes
1answer
27 views

Proof of Vector Space Axioms [on hold]

Where can I find detailed proof of vector space axioms? Any reference to a book, website or video lecture.
1
vote
0answers
8 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...
3
votes
0answers
80 views

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$ where $S$ is a invertible matrix and $B$ has the form $B = \left( \begin{array}{ccc} 0 & a_1 & 0 & ...
1
vote
0answers
33 views

Proving that elementary row operations are preserved after multiplication

If $E$ is an elementary $n \times n$-matrix, show that if $A$ is any $n\times n$-matrix, then $EA$ is a matrix obtained by carrying out a single elementary row operation on $A$, and that $AE$ is a ...