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

1
vote
1answer
20 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
27 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
1answer
19 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
38 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
18 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
53 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
25 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
30 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
35 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
24 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
61 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
99 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
33 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
10 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
42 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
29 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
1answer
22 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
39 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
19 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
12 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
42 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
35 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
10 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
105 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
37 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 ...
0
votes
1answer
12 views

If we have a singular matrix, what can we say about it's adjoint [on hold]

We need to prove that for singular matrix determinant of its adjoint is also 0.
2
votes
1answer
43 views

Proof that an involutory matrix has eigenvalues 1,-1

I'm trying to prove that an involutory matrix (a matrix where $A=A^{-1}$) has only eigenvalues $\pm 1$. I've been able to prove that $det(A) = \pm 1$, but that only shows that the product of the ...
-4
votes
0answers
25 views

Do prove in vector space (about span and subspace)

(a) let vector u = (a,b,b,0) and vector v = (0,c,-c,d) because u dot v = 0, thus v = c(0,1,-1,0) +d (0,0,0,1) therefore, W2 = span {(0,1,-1,0),(0,0,0,1)} the basis for w2 is (0,1,-1,0),(0,0,0,1) for ...
0
votes
0answers
54 views

Formula for Area of parallelogram induced by linear operator

I'm given that the linear operator $L: \mathbb R^2\to\mathbb R^2$ is invertible. The set (u,v) is a linearly independent set in $\mathbb R^2$. I must find a formula for the area of the parallelogram ...
0
votes
2answers
17 views

Find a Cartesian Equation for the Plane Satisfying Those Properties

Find the Cartesian equation of the plan parallel to j and passes through the intersection of the planes described by the equations x + 2y + 3z = 4, and 2x + y + z = 2. I was able to get the ...
0
votes
1answer
39 views

If $A$ is skew-symmetric, then a fixed row/column operation produces a new skew-symmetric matrix

Suppose $A$ is a skew-symmetric matrix. Fix an elementary row operation. If we carry out this row operation on $A$, and then carry out the corresponding column operation on the resulting matrix, do we ...
2
votes
1answer
12 views

Possible values of nullity in 4x2 matrix

Let $A$ be a 4 by 2 matrix. Explain why the rows of $A$ must be linearly dependent. What are the possible values of nullity(A)? I understand the first part. I do not understand the second part. The ...
1
vote
0answers
12 views

Given a singular covariance matrix of a random vector $X=[X1,X2,X3,X4]$, how do I partition $X$ into 2 parts to get a non-singular covariance matrix?

Suppose I have a covariance matrix of a random vector $$X=\begin{bmatrix}X1\\ X2\\ X3\\ X4 \end{bmatrix}$$ $$C_X=\begin{bmatrix}a&b&c&d\\ e&f&g&h\\ i&j&k&l\\ ...
2
votes
1answer
47 views

Show that $EA$ is obtained from an elementary row operation on $A$

Suppose $E$ is an elementary $n \times n$-matrix. Prove that if $A$ is any $n\times n$-matrix and $E$ is any elementary matrix, then $EA$ is a matrix obtained by carrying out a single elementary row ...
0
votes
2answers
42 views

To find Basis and kernel of matrix A

Given a matrix $A:$ \begin{pmatrix} 1 & 2 & 5 \\ 3 & 5 & 13 \\ -2 & -1 & -4 \end{pmatrix} My textbook has reduced it to RREF to find kernel and dimension of it. To find the ...
1
vote
1answer
36 views

An inequality about Hermitian matrices

Say one knows the following statement, That for any Hermitian matrix $H$ with eigenvalues $\lambda_1 \geq \lambda_2 ..\geq \lambda_n$ one has, that in any basis, for any positive integers $1 \leq i_1 ...
1
vote
1answer
53 views

Decompose $\omega:= e_0\wedge(e_1\wedge e_2 + e_3\wedge e_4)$

$(e_1\wedge e_2 + e_3\wedge e_4)$ is well-known to be of rank 2 (can't be decomposed). On the other hand, $\omega \wedge \omega = e_1\wedge e_1 \wedge ... = 0$. According to the wikipedia article ...
0
votes
1answer
43 views

Show that the set $\{v_n\} \subset l^2$, $v_j$ is orthonormal.

Show that the set $\{v_n\} \subset l^2$, where $v_n=\frac{1}{\sqrt{2}}(e_n-e_{n+1})$ if $n$ is odd and $v_n=\frac{1}{\sqrt{2}}(e_n+e_{n-1})$ if $n$ even is orthonormal. Is it a Schauder basis of ...
0
votes
1answer
22 views

Tricky change-of-basis transformation problem

I have absolutely no idea what to do here because of the $\sin(x).$ Let $V = \text{Span}\left\{x, x^3, \sin(x) \right\}$, and consider the basis for $V$ given by $\beta = \left\{x-2x^3, x^3+\sin(x), ...
0
votes
1answer
15 views

Necessary and sufficient conditions for an adjoint of a linear map to be the map's inverse

Let $V$ be a finite dimensional inner product space, $ \phi :V \rightarrow V$ a linear operator and $\phi^*:V \rightarrow V $ its adjoint. I wish to show: $\phi^*$ is an inverse to $\phi$ if and ...
1
vote
1answer
40 views

Can -3 and 2 be eigenvalues of the following matrix?

Can $-3$ and $2$ be eigenvalues of and nxn matrix B such that $A = B^{2}+B-6I$ and A's determinant is $0$? So this is what I concluded: At first glance, it can be seen that the matrix $A$ can be ...
0
votes
0answers
16 views

How to show that $F(m_0+v)=m_0+A(v), v∈V$ defines an affine map of $(M,V)$? [on hold]

Let$(M,V)$ be an affine space, and let $m_0 ∈ M, A ∈ L(V)$. I need to show that the equation $$F(m_0+v)=m_0+A(v), v∈V$$ defines an affine map of $(M,V)$ with linear part $LF = A$ that has $m_0$ as a ...
0
votes
2answers
30 views

Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis ...
2
votes
0answers
37 views

Shortlist of problems in linear algebra

A while ago I remember seeing a very nice shortlist of problems in linear algebra. It was a list of about 40-50 problems. The idea was that if you solve them, you learn linear algebra very well and ...
2
votes
2answers
36 views

Need some help on linear algebra Subspace test

Any help would be appreciated, thank you.
1
vote
0answers
37 views

Eigenvalues of a matrix with special form

Let $p,a_1,...,a_n\in(0,1)$ and $\sum_{i=1}^na_i=1$. Now consider the following matrix: $$ \left(\begin{array}{ccccc} (1-p) & \sqrt{p(1-p)}a_1 & \sqrt{p(1-p)}a_2 & ... & ...
1
vote
2answers
40 views

permutation matrix?

$\sigma=(132)(45)$ be a permutation, we construct the matrix $A$ such that its $i'$th collumn is the $\sigma(i)'$th colllumn of $I_5$. so the matrix which I got is ${A}=\left[\begin{array}{*{20}{c}} ...
0
votes
2answers
19 views

Computing orthogonal projection

The question asks: A vector u and a line L in R^2 are given, compute the orthogonal projection w of u on L. u=[3,4] and y=-x In one example i was given two ...