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 ...

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12
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1answer
245 views

Variety of pairs of product-zero matrices

Here's an old qualifying exam question I got stuck on. Consider the variety $X$ of pairs of matrices $(A,B)$ satisfying $AB = BA = 0$ (with entries in some field). What are the irreducible components ...
3
votes
1answer
322 views

Properties of different matrix representations of clifford algebras

I am looking for some theorems about matrix representations of Clifford algebras. Let $a \in G_{p,q,r}$, where $p$ elements square to $1$, $q$ to $-1$, and $r$ to $0$, that is ...
1
vote
1answer
193 views

Kalman filter and data extrapolation

Context of the situation: I have a system set up that can give me the position of a person in a room. I also have a light that shines on this position. However, the light are lagging behind by 0.300 ...
10
votes
2answers
511 views

Do determinants of binary matrices form a set of consecutive numbers?

While pondering a solution for the problem of generating random 0-1 matrices with small absolute determinants, I once again realise how little I know about 0-1 matrices. My initial idea was to pick a ...
3
votes
0answers
359 views

Trace Minimization of Covariance Matrix

Given a matrix X whose rows contain observations collected at some locations. Can someone explain how trace minimization of covariance matrix $XX^T$ can lead to orthogonal / mutually independent ...
3
votes
1answer
424 views

Cosine similarity between complex vectors

Having a pair of vectors where its elements are complex, is that possible to calculate the cosine similarity between these two vectors to see how similar they are?, and if the result of this ...
5
votes
1answer
147 views

Linear Algebra Text Problem

We have specified number of light bulbs. In addition to the array there are buttons. Pressing the button changing state of light bulbs which are connected to the switch. It is known that for each set ...
3
votes
2answers
65 views

rank of specified matrix

I have a task to prove, that: $$\mathrm{rank} \left(\begin{bmatrix}A &A^2 \\ A^3 & A^4 \end{bmatrix}\right)=\mathrm{rank}(A)$$ Please give me an advice. Is ...
3
votes
1answer
97 views

Echelon form of matrix have the same rank, after operations modulo

In my homework task, I need to prove that if my matrix have rank $k$ modulo $\ell$, then it also have rank $k$ modulo $p$. Please give me advice, how to prove it. --edit-- The precise task is: On ...
3
votes
1answer
159 views

Outward vectors to an Ellipsoid and Euclidean metrics

I'm reading Arnold's proof of the topologically equivalence of the equations $\dot{x}=Ax$ and $\dot{x}=x$ when all the eigenvalues of the $n \times n$-matrix $A$ have positive real part. The proof is ...
1
vote
3answers
136 views

Prove there is a homogeneous system of equations where solution is $M \subseteq \mathbb{R}^n$

So, I've been thinking of how to prove this. For example. Let $\{(1, 1, 0, -1), (1, 0, 1, 2)\}$ be the base vectors of subspace $M \subseteq \mathbb{R}^4$. One needs to show there is a system of ...
1
vote
1answer
54 views

Linear transform of parameterization, but implicit?

Suppose I have a $2\times 2$ matrix $M$ in implicit equation form, where $(x,y)$ is transformed to $( xOut,yOut )$: $$ ax + by = xOut, \quad cx + dy = yOut $$ Now additionally I have a line $L$, also ...
4
votes
2answers
127 views

Where have I made my mistake in calculating $P^{-1}AP$?

I have $$Q = \begin{pmatrix} -\mu & \mu \\ \lambda & -\lambda \end{pmatrix}$$ and I want to work out the value of $\mathbb{P}(t) = \exp(Qt)$ So I diagonalised $Q$ and then worked out the ...
0
votes
1answer
55 views

$\text{Im}(S+T) \subseteq \text{Im}(S) + \text{Im}(T)$

We are given two linear transformations - $S,T : V \to V$. how do I prove that $\text{Im}(S+T) \subseteq \text{Im}(S) + \text{Im}(T)$?
1
vote
1answer
106 views

For a first order inhomogenous system of linear differential equations, what is a good way of defining resonance?

I apologize for the title being slightly unclear (at least to me it seems so), so if anyone has a better suggestion feel free to change it. Anyways, for example, when dealing with a second order ...
3
votes
3answers
596 views

How to randomly construct a square full-ranked matrix with low determinant?

How to randomly construct a square (1000*1000) full-ranked matrix with low determinant? I have tried the following method, but it failed. In MATLAB, I just use: n=100; A=randi([0 1], n, n); while ...
3
votes
1answer
1k views

How to find Matrix Inverse over finite field?

How to find matrix Inverse over finite field? I am using MATLAB, and I know gf() in MATLAB can enable me to do linear algebra operations over finite field $F_{2^m}$ for some m. However, if I want to ...
1
vote
1answer
98 views

How to go from the diagonal matrix back to the matrix with the original basis?

I start off with matrix $Q$, such that $$Q = \begin{pmatrix} -\mu & \mu \\ \lambda & -\lambda \end{pmatrix}$$ and I want to work out the value of $P(t) = \exp(Qt)$ So to do this, you first ...
0
votes
1answer
141 views

Inequality for singular value for differences of matrices (upper bound)

Does anybody know the inequality of singular value for differences of matrices, i.e. $\sigma_{max}\left(\begin{array}{c} A-B\end{array}\right)\leq??$ in term of $\sigma_{max}\left(\begin{array}{c} ...
5
votes
1answer
91 views

Is it possible to reverse this sequence of permutations?

Let $ S = (a_1, a_2, ..., a_N) $ be a finite (arbitrarily long) sequence of elements, and let $p_1, p_2, ..., p_n $ be the first $n$ prime numbers, with $n \ge 3$. We apply a sequence of ...
0
votes
2answers
66 views

SVD - can the non-zero-singular-values be all lower than 1?

Let $$X \in \mathbb{R}^{n \times m}$$ be a (not-zero) matrix. Consider its Singular Value Decomposition $$X = U\Sigma V^T = U[\text{diag}(\sigma_1, \sigma_2, \dots, \sigma_k )]V^T$$ Is it possible ...
1
vote
1answer
1k views

How to find the intersection of these two points graphically?

I have the following equations : 1) $3x-y=4$ 2) $4x-2y=2$ Now on paper, I did the following : 1) at $x=0$, $y=-4$ , at $y=0$, $x=1.333$ 2) at $x=0$, $y=-1$ , at $y=0$, $x=\frac{1}{2}$ Using ...
3
votes
1answer
441 views

Find basis of im, ker and dim im, dim ker verification

In my homework, I've to find basis of im and ker of linear transformation $\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}, \varphi((x_{1},x_{2},x_{3}))=(2x_{1}+x_{2}-3x_{3},x_{1}+4x_{2}+2x_{3})$ My ...
0
votes
1answer
41 views

A set of Linear transformations

Let $$S = \{T:\mathbb{R}^3\to\mathbb{R}^3:T\text{ is a linear transformation with }T(1,0,1)=(1,2,3),T(1,2,3)=(1,0,1)\}$$ Then S is: (a) a singleton set (b) a finite set containing more than one ...
0
votes
1answer
52 views

Order of matrices

If $n$ is the least positive integer such that: $$\begin{pmatrix} \cos\frac{\pi}{4} & \sin \frac{\pi}{4}\\ -\sin\frac{\pi}{4} & \cos\frac{\pi}{4}\end{pmatrix}^n$$ is the identity ...
0
votes
1answer
82 views

Kernel of linear combination of functionals

Given linear functionals $\varphi_2 = ( 2 , -2\alpha, 1)$ and $\varphi_1 = (2,1 -\alpha, -1)$ how do I find kernel of $\varphi_1 - 2\varphi_2$? ($\alpha$ is some parameter) Is it so simple as ...
5
votes
2answers
141 views

Calculate $\left\Vert \begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} \right\Vert$

With $$\left\Vert A \right\Vert=\max_{\mathbf{x}\ne 0}\frac{\left\Vert A\mathbf{x}\right\Vert }{\left\Vert \mathbf{x}\right\Vert }$$ and $$A=\begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} $$ ...
1
vote
2answers
91 views

smallest eigenspace dimension symmetric

Hi would you help me with the following: Let $A = (a_{ij}) R^{n \times n}$ be a symmetric matrix satisfying: $a_{1i} \neq 0$; Sum of each row equals $0$ and each diagonal element is the sum of ...
2
votes
2answers
40 views

How can I recover a variable with singular coefficent

How can I recover $X$ from \begin{equation} AX = B \end{equation} when $A$ is a singular matrices?
2
votes
5answers
207 views

Calculation matrix exponential

I got $$ A = \begin{pmatrix} 0 & \omega \\ - \omega & 0 \end{pmatrix}$$ with eigenvalues $\pm i\omega$ and eigenvectors $(-i,1)$ and $(i,1)$. Can I then calculate $e^{tA}$ by $$ e^{tA} = V ...
2
votes
2answers
64 views

Splitting a tensor

Is it possible to write $$\int d^3x \,\,\, x_i\,\,x_j\,\,\,f(\vec x)$$ where $f(\vec x)$ is some function of the position and the indices indicate which component, as a sum of a traceless tensor and ...
3
votes
1answer
65 views

Matrix of linear transformation format $(m,n)$

I'm having a problem with doing an exercise in construction of a matrix by a given linear transformation. So I understand ex. $f(x,y) = (2x, -y)$ is \begin{matrix} \\ 1 & 0\\ ...
4
votes
1answer
75 views

Distance is independent of coordinates

I am asked to show $d(x,y) = ((x_2 - x_1)^2 + (y_2 -y_1)^2)^{1/2}$ does not depend on the choice of coordinates. My try is: $V$ has basis $B = b_1 , b_2$ and $B' = b_1' , b_2'$ and $T = [[a c], [b ...
0
votes
1answer
103 views

How to Simplify this to the given answer? (matrix equation, trig functions)

Can someone help me to simplify $$\Bigg( \begin{matrix} 5\cos t &5\sin t \\2\cos t+\sin t & 2\sin t-\cos t\end{matrix} \Bigg) \Bigg( \begin{matrix} u_1 \\u_2\end{matrix} \Bigg) ...
3
votes
1answer
61 views

Truncation of the projection of V - isomorphism

a) Let $W$ and $Z$ be two vector subspaces of $V$ and $W\cap Z=\{0\}$. Prove that every isomorphism from $W$ to $Z$ is the restriction of a projection operator defined on $V$. b) Prove that if ...
3
votes
1answer
77 views

Dim E, set of linear transformations

Suppose $U\subset W \subset V$ are three linear spaces with respective dimensions 3, 6 and 10. Let $E\subset L(V,V)$ be the set of linear transformations $f:V\rightarrow V$ such that $f(U)\subset U$ ...
0
votes
1answer
223 views

What is a basis for the space of anti-symmetric $3\times 3$ matrices?

I tried to find a basis for the subspace of 3-by-3 anti-symmetric matrices - but for nothing. How to find such a basis?
7
votes
2answers
350 views

Maximal dimension of subspace of matrices whose products have zero trace

In the space of all real matrices with dimension $n$, find the maximal possible dimension of a subspace $V$ such that $\forall X,Y\in V,\, \operatorname{tr}(XY)=0$.
3
votes
3answers
511 views

Rank of matrix AB when A and B have full rank

Define $A$ as $m\times n$ matrix with rank $n$, and $B$ as $n\times p$ matrix with rank $p$. Calculate the rank of matrix $C=AB$. --edit-- Rank of a matrix is the number of linear independent rows.
3
votes
1answer
130 views

If $AB=0$, then $A+A^T$ or $B+B^T$ is singular

Define $A$ and $B$ as being square matrices of dimension $2011$. Prove that if $AB=0$, then at least one of matrices $A+A^{T}$ or $B+B^{T}$ have rank below $2011$. -- edit -- Rank of a matrix is ...
2
votes
2answers
311 views

Linear Algebra- independence and dependence

Let $u$ and $v$ be non-zero vectors in $V$. Prove or disprove the following claim. $u$ and $v$ are linearly dependent $\implies$ $(u+v)$ and $(u-v)$ are linearly dependent. Is the following proof ...
3
votes
1answer
546 views

Visualizing the four subspaces of a matrix

Given a system of linear equations in the form $$AX=b$$ How can I go about visualizing the four fundamental sub-spaces - column space, row space, null space and left null space? In the same context, ...
2
votes
2answers
210 views

A problem on self adjoint matrix and its eigenvalues

Let $S = \{\lambda_1, \ldots , \lambda_n\}$ be an ordered set of $n$ real numbers, not all equal, but not all necessarily distinct. Pick out the true statements: a. There exists an $n × n$ matrix ...
0
votes
1answer
364 views

a multiple choice question on symmetric matrices [duplicate]

Possible Duplicate: real symmetric matrix $B=AA^{*}$? Let $A$ be an $n×n$ matrix with real entries. Pick out the true statements: a. There exists a real symmetric $n × n$ matrix $B$ such ...
2
votes
3answers
910 views

If $\lambda^k$ is an eigenvalue of $A^k$, is $\lambda$ an eigenvalue of $A$?

The converse is easy to show by multiplying $Ax$ on the left by $A$ (x is a eigenvector): $AAx=A(\lambda x)=\lambda Ax=\lambda^2x$ but I was wondering if the converse is true. The same approach ...
4
votes
3answers
223 views

Optimal symmetric rank-1 approximation

I want to find $\mathbf{x}$ that minimizes $\|A-\mathbf{x}\mathbf{x}'\|^2$ where $\|\cdot\|$ is Frobenius norm. Differentiating with respect to $\mathbf{x}$ and setting to $\mathbf{0}$, I get ...
30
votes
3answers
1k views

Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
7
votes
1answer
278 views

Prove that if $A^2x=x$ then $Ax=x$

I feel this should be easy but I cant solve this problem: Prove that if $A$ is a $n\times n$ matrix and $x$ a vector in $\mathbb R^n$ both with real positive entries and $A^2x=x$ then $Ax=x$. I ...
1
vote
1answer
52 views

Nullspace as a subspace of $\mathbb R^n$

Why is the nullspace of an $m\times n$ matrix $A$ a subspace of $\mathbb R^n$ whereas the column space is a subspace of $\mathbb R^m$? I understand the dimension of $C(A)$ is designated by the number ...
4
votes
1answer
85 views

Angle after two rotations in $\mathbb R ^3$

Question: A rotation through $45^{\circ}$ about the x-axis is followed by a similar one about the z-axis. Show that the rotation corresponding to their combined effect has its axis inclined at equal ...