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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
45 views

If dim V = dim U and $S \circ T$ is onto, prove or disprove V and W are isomorphic

Let $T: \mathbb V \to \mathbb W$ and $S: \mathbb W \to \mathbb U$ be linear maps. If dim $\mathbb V$= dim $\mathbb U$ and (S o T) is onto (composition), then $\mathbb V$ and $\mathbb W$ are ...
0
votes
0answers
12 views

Prove that eigenvalues of the operator lie in the interval [on hold]

Let $\phi$ and $\psi$ be two self-adjoint linear operator in Euclidean space, the eigenvalues of which lie respectively in the intervals $[c; d]$ and $[m; n]$. Prove that eigenvalues of the operator ...
1
vote
0answers
18 views

Subtracting scaled projection matrix from identity matrix

I am trying to understand what the following operation signifies. $$ \rm W_n=I-2u_n u_n^H/u_n^Hu_n $$ where I and $u_n$ is described in section 6.3.4.2.3 of this document. My question is, what does ...
0
votes
0answers
18 views

Let $S$ be a set of vectors. If each finite subset of $S$ is linearly independent, then $S$ is linearly independent

Here's a similar question which doesn't answer mine: If every subset of $S$ is linearly independent, then $S$ is independent Let $S$ be finite. Let $S_1 \cup S_2 \cup \ldots \cup S_n = S$ such that ...
1
vote
1answer
105 views

Eigenvectors of “weighted” Hermitian matrix?

Consider two real matrices $\boldsymbol{H}$ and $\boldsymbol{D}$ with the following properties: $\boldsymbol{H}$ is a symmetric matrix (since it is a real matrix this is equivalent to being ...
4
votes
1answer
78 views

Can a elementary row operation change the size of a matrix?

My question is very simple - Can an elementary row operation change the size (eg: $2\times2$ or $3\times 2$) of a matrix? I think the answer should be no, but while reading Linear Algebra by Hoffman ...
2
votes
1answer
51 views

Lie bracket of $\mathfrak{so}(3)$

I know that for $\mathfrak{so}(3)=\mathcal{L}(SO(3))$, the set of $3\times 3$ real antisymmetric matrices, we can define a basis $$T^1=\begin{pmatrix}0&0&0\\ 0&0&-1\\ ...
0
votes
1answer
29 views

Systems of linear equations in the same modulus

I am working with a system of linear equations all taken with the same modulus, $n$, there is no assumption on $n$ other then it is at least 3 (really don't want to assume it is prime) I don't have ...
0
votes
3answers
54 views

Express eigenvectors of $A^{-1}$ in terms of eigenvectors of $A$

I know the eigenvalues of the matrix $A^{-1}$ are $\frac{1}{\lambda_n}$ where $\lambda_n$ are the eigenvalues of $A$. I didn't know their eigenvectors were related; in what way are they related? Also ...
1
vote
1answer
42 views

Degrees of freedom in a $n \times n$ table

Suppose we have an $n \times n$ table where each row and each column sums to some number $k$. Say that the elements of the table and $k$ are real numbers. Now the question is how many places can we ...
0
votes
1answer
23 views

finding inner product

This is from my textbook: I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, ...
1
vote
0answers
20 views

Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
0
votes
1answer
17 views

Elementary reflector $Q$ is orthogonal iff

Recall that an elementary reflector has the form $Q = I + \alpha xx^T\in\mathbb{R}^{n\times n}$ with $\|x\|_{2}\neq 0$. Show that $Q$ is orthogonal iff $$\alpha = \frac{-2}{x^Tx} \ \ \text{or} \ \ ...
4
votes
3answers
515 views

Can a matrix satisfy all three of the following properties?

Consider an $n \times n$ matrix of the form $$ A = \begin{bmatrix} a_1 & a_2 & \ldots & a_{n-1} & a_n \\ 1 \\ & 1 \\ & & \ddots \\ & & & 1 \end{bmatrix} $$ for ...
1
vote
3answers
36 views

Irreducible polynomials in $\mathbb F_3[x]$

Finding irreducible polynomials in $\mathbb F_3[x]$ of degree less or equal to $4$ for $d=2,3$ the polynomial should not have a root case $d=2$ there are $2\cdot 3\cdot 3=18$ polynomials with ...
4
votes
0answers
42 views

How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left ...
-1
votes
1answer
16 views

Is a subset of a linearly independent set in a vector space linearly independent?

The empty set is linearly independent. Also, $S_1$ is linearly independent if $S_1 = S_2.$ Suppose $v_1, \ldots, v_n \in S_2.$ Then $a_i = 0$ for all $i$ in $a_1v_1 + \ldots + a_nv_n = 0.$ Now ...
1
vote
1answer
58 views

Square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue

Prove square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue. My proof: If matrix has eigenvalue z=$\frac{1}{4}(-3+ i \sqrt5)$, then it must has eigenvalue ...
1
vote
1answer
25 views

Prove that the dimension of row space equals to the dimension of column space of an $n\times n$ matrix

Knowing that the row space of $A\in \mathbb{R}^{n\times n}$ equals $N(A)^\perp$ prove that the dimension of column space of a matrix equals its row space dimension. So I'm trying to apply ...
0
votes
0answers
18 views

How many ordered basis does $V$ have? [duplicate]

If $F$ is a field with $q$ elements, $V$ a $F$-vector space of dimension $n$, then how many ordered basis does $V$ have ? First, $V$ has $q^n$ elements, am I correct ? Let $\{v_1,\dots,v_n\}$ be ...
0
votes
0answers
34 views

Finding bases for the kernel and image of $M= \begin{bmatrix}1&1&1\\-s&2+s&-1\\-s&s&1\end{bmatrix}$

So I have a Matrix $$M= \begin{bmatrix}1&1&1\\-s&2+s&-1\\-s&s&1\end{bmatrix}$$ and a linear transformation $$f:\mathbb{R}^3 \to \mathbb{R}^3, \quad x \mapsto Mx$$ and I'm ...
0
votes
0answers
27 views

Norm of a dot product?

I am reading a paper and is rather perplexed by the following equation. Particularly, inside the double bar || which I believe is the norm, there is a dot product. If that is the case, what does it ...
1
vote
3answers
46 views

How can I prove: $\forall \lambda \in C$ with $|\lambda| = 1$, exist unitarian matrix $B$ with eigenvalue $\lambda$

How can I prove: $\forall \lambda \in C$ with $|\lambda| = 1$, exist unitarian matrix $B$ with eigenvalue $\lambda$. I tried to find a counter-example and I was not succeeded. I believe I need to ...
0
votes
1answer
23 views

Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and ...
0
votes
1answer
18 views

Given $f(x)=x^2-4x+3$, find the points on the curve $y=f(x)$ where the tangent to the curve passes through -6.

Given $f(x)=x^2-4x+3$, find the points on the curve $y=f(x)$ where the tangent to the curve passes through $(0,-6)$. State the equations of the tangents at these points. Hi everyone, I tried to find ...
0
votes
0answers
14 views

Division vector space over binary extension

I'm working in the maths behind the Rijndael's algorithm and I get stalled with the mixColumns() operation. I'm writing code to do the algorithms without pre-calculation tables (the objective is to ...
1
vote
1answer
35 views

Existence of Unimodular Congruence Transformation for Symmetric, Integer matrices

Two symmetric, integer valued matrices, $K_1$ and $K_2$, are congruent if there exists a unimodular integer matrix, $X$, such that $$X^T K_1 X = K_2$$ What are the conditions on the existence of such ...
0
votes
0answers
20 views

Rank of a symmetric matrix after removing a column and row.

If I have a $n\times n$ symmetric matrix $M$ with real entries, zeros on the diagonal, and two of the column vectors are identical and I remove one of these columns, and the corresponding row, then ...
0
votes
0answers
18 views

Query about the Moore Penrose pseudoinverse method

I have recently discovered the Moore-Penrose psuedoinverse method, and I am currently testing the waters with it. I noticed if I have a system, say $$a_1x_1=0$$ $$a_2x_1+a_3x_2=0$$ $$\vdots$$ ...
1
vote
1answer
33 views

Effect of simple linear transformation

Consider the linear transformation given by $$T\left\{\begin{bmatrix}x \\y\\z\end{bmatrix}\right\} =\begin{bmatrix}-x\\y\\z\end{bmatrix}$$ Find a matrix $A$ such that $T(x) = Ax$, where x = $[x, y, ...
3
votes
0answers
29 views

Eigenvectors of Generalized Sylvester Equation $AX+XB^\text{T}=\lambda CXD^\text{T}$

Ok here's what I mean with the Sylvester equation eigenvectors. The simplest case, where $C = D = I$, has already been solved in the literature (Matrix Calculus by W.H. Steeb). $$A X + X ...
-1
votes
0answers
19 views

PTM using Hastings-metropolis [on hold]

[Compute the 4 × 4 PTM (pij ) under the T = 2 dynamics of Hastings–Metropolis][1]
0
votes
2answers
23 views

Find the possible equations of planes containing a line

If I had a line in $R^3$ that had all its points described by $P =\begin{pmatrix} a\\ b\\ c \end{pmatrix}+\lambda \begin{pmatrix} x\\ y\\z \end{pmatrix}$ where $a, b,$ and $c$ are constants, what ...
0
votes
0answers
47 views

Does $B^2 \leq A^2$ imply $\| A^{-1} B\| \leq 1$ for the operator norm?

Assume we have two $n \times n$ real symmetric matrices $ A^2 $ and $B^2$, such that it holds for some $0\leq\rho<1$ $$ 0 < (1-\rho)B^2 \leq A^2 \leq (1+\rho)B^2, $$ where "$\leq$" means ...
0
votes
1answer
18 views

Number of vectors in a span over a certain field

Let $S = \{u_1, u_2,\ldots, u_n\}$ be linearly independent subset of vector space $V$ over $\mathbb Z_2$. Number of vectors in $\operatorname{span}(S)$? Consider $u_1 + u_2 + \cdots + u_n \in ...
0
votes
0answers
50 views

Is there any linear algebra textbook presented using logical symbols?

I'm currently going through a book called Linear Algebra Done Right by Axler, and to be honest, his book seems to be very loose with what things he defines. For instance , the symbol 0 could be mean a ...
5
votes
1answer
91 views
+100

Given a symmetric positive-definite matrix $M$, find all $A$ such that $A^\top M A=M$

Given $M$ a real symmetric positive-definite matrix, I would like to characterise all matrices $A$ such that $A^\top M A=M$. Note that the question of finding $A$ solutions to $A^\top M A=M$ for all ...
1
vote
1answer
34 views

Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So ...
1
vote
0answers
15 views

Finite prime field representation of uniform matroid $U_{2,n}$

Suppose I have a uniform matroid $U_{2,n} = (E, I)$ (so $F \subset E$ has $F \in I \iff |F| \leq 2$) and want to represent it over $GF(p)$, i.e. I would like to construct a map $\phi : E \to GF(p)^2$ ...
1
vote
2answers
31 views

How do I specify the inverse of a correlation matrix?

To specify a correlation matrix $\in \mathbf{R}^{n\times n}$. There are $n(n-1)/2$ free elements. If I wanted to specify a matrix that is the inverse of some correlation matrix, how should I specify ...
0
votes
0answers
10 views

Let $C\in\mathbb{R}^{p\times n},~A\in\mathbb{R}^{n\times n}$. When does $[C; CA^m]$ have rank $n$?

Let $C\in\mathbb{R}^{p\times n},~A\in\mathbb{R}^{n\times n}$, where $A$ is a nonsigular matrix. Define the following set of $\bar{m}$ matrices: $$O_m = \begin{bmatrix} C \\ CA^m ...
0
votes
1answer
33 views

Square of any determinant is symmetric.

This property is given in my book. The square of any determinant is a symmetric determinant. Well it works when I take a determinant say $3 \times 3$ and multiply it by itself using row to row ...
1
vote
5answers
47 views

Linear independence of a set of $4$ polynomials

Determine whether the next set of vectors $$\{x^2-1, 2x-3, x^2+1, 4x\}$$ is a linearly independent or a non linearly independent subset of $P$ (the vector space of polynomials). It was today in ...
4
votes
1answer
79 views

When does a matrix admit a Jordan canonical form?

If a matrix over the field $\mathbb R$ has as elementary divisors: $x-4$, $x^2 + 2$, does it then admit a Jordan canonical form? Am I right in thinking that a matrix has a Jordan canonical form ...
1
vote
3answers
47 views

Finding the matrix for a linear transformation on a vector space when the basis changes

Let B={$u_1,u_2,u_3$} as basis of Vector Space V, and Let T: V→V be the linear operator defined by, $$ [T]_B=\begin{bmatrix} -3 & 4 & 7 \\ 1 & 0 & -2 \\ ...
1
vote
0answers
70 views
+500

Weyl group, bilinear form, and character/cocharacter pairing. Many questions!

Let $G$ be a connected linear algebraic group, $T$ a maximal torus of $G$, and $\alpha$ a weight of $T$ such that $G_{\alpha} = Z_G(S)$ is not solvable, where $S = (\textrm{Ker } \alpha)^0$. I have ...
2
votes
1answer
17 views

Diagonalizability and elementary divisors

How to prove that an $n \times n$ matrix $A$ over a field $\mathbb F$ is diagonzalizable if and only if every elementary divisor of $A$ has degree $1$? I kind of know why this is true but I am not ...
0
votes
1answer
17 views

Finding Transition Probabilities using Metropolis Hastings

I want to find the $4$x$4$ Probability Transition Matrix under the temperature parameter T=2 of Metropolis Hastings. I know that, if x and y are neighbors, $p(x,y) =$ $$ f(x) = \left\{ ...
1
vote
1answer
24 views

Lower bound for norm of matrix

I have the following problem: $A$ is a positive definite, symmetric matrix. Firstly I was required to find a matrix $B$ such that $B^n = A$. I believe this to be $C(D^{\frac1n}) C'$ where C is the ...
0
votes
0answers
26 views

If some columns of $XA, A$ are equal, does it mean $XA=A$?

I'm working on a problem related to the row space $R(A)$ of a matrix $A \in K^{k \times n}$, where $k < n$. This space is invariant under a left-action of $GL(k, K)$ on the matrix $A$. Say I have ...