For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

learn more… | top users | synonyms (2)

0
votes
0answers
48 views

Comparing Bases in $\mathbb{R}^{n}$

A bit of trouble with the following question: Let $\mathcal{B}$ be the basis of $\mathbb{R}^{n}$ consisting of the vectors $\vec{v_1},\vec{v_2},\cdots,\vec{v_n}$, and let $\mathcal{E}$ be some other ...
6
votes
3answers
278 views

Square root-related calculations with matrices

If $\mathbf A$ is an $n \times n$ matrix such that $\mathbf A^6 = \mathbf I_n$ (the identity matrix), is it true that either $\mathbf A^3 = \mathbf I_n$ or $\mathbf A^3 = \mathbf -I_n$? I'm ...
0
votes
0answers
79 views

Equivalence proofs concerning matrices and systems of linear equations

Let $\mathbf A$ be the augmented $m × (n + 1)$ matrix of a system of $m$ linear equations with $n$ unknowns. Let $\mathbf B$ be the $m × n$ matrix obtained from $\mathbf A$ by removing the last ...
0
votes
1answer
33 views

Is the observed widest-width of an oblate sphere constant under all rotations?

This is something which I feel intuitively is true but I'm having trouble finding a way of proving it mathematically. Given an oblate sphere, or ellipsoid, with equation $$x^2+y^2+(z^2 / c^2)=1, c<...
0
votes
1answer
523 views

Constructing the Matrix B “column by column”

I'm going through the various ways to construct a B-matrix of a linear transformation and I'm hitting a snag with one of the methods. We have $A = \begin{pmatrix} -3 & 4 \\ 4 & 3 \\ \end{...
0
votes
1answer
54 views

FP3 Vectors question

$$\mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}$$ $$\mathbf{b}=b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k}$$ $$\mathbf{c}=c_1\mathbf{i}+c_2\mathbf{j}+c_3\mathbf{k}$$ Use appropriate determinants ...
2
votes
2answers
69 views

Show an integral of continuous $\gamma:\mathbb{R}\to GL(n,\mathbb{R})$ is invertible.

Let $$\gamma:\mathbb{R}\to GL(n,\mathbb{R})$$ be a continuous map such that $$\gamma(s+t)=\gamma(s)\gamma(t),\quad\gamma(0)=I,$$ for all $s,t\in\mathbb{R}$, and let $\psi:\mathbb{R}\to[0,\...
0
votes
1answer
62 views

Symmetric matrix-Spectral theorem

Assume we have a matrix $A$ let's say $100 \times 4$. We determine the product $B=A^{T}A$ Then by the spectral theorem \begin{equation} B =U^{T} \lambda U \end{equation} $B$ is a symmetric matrix $...
0
votes
1answer
30 views

How is this norm $\|A\|_*$ called and what is it?

In my lecture notes I have found the notation $\|A\|_*$ for a matrix norm. Do you know the name of this norm (such that I can read the definition of it), or do you even know the definition of it? ...
1
vote
1answer
87 views

Square matrices A and B commute if and only if they share the same generalized eigenspace.

I found a couple of proofs for this theorem but only for the case when A and B are diagonalizable, thus the eigenspace that they share is not the generalized one. Im looking for the proof (or ...
0
votes
1answer
18 views

How can we use the symmetry of this complex matrix?

Find the Jordan normal form of $A\in \mathbb C^{4,4}$ if A is symmetric, $A^2=A$ and $\operatorname{rank} A=3$. So $A^2=A$ implies that the only eigenvalues are $0$ and $1$. From $\operatorname{rank}...
0
votes
3answers
253 views

No two 2x2 matrices in Jordan form are similar?

Let $S$ be the set of 2x2 matrices in Jordan Normal form $\begin{pmatrix}x&a\\ 0&y\end{pmatrix}$ with $a=0$ or $1$ and $x \leq y$. How do I show that no two matrices in $S$ are similar? Thank ...
0
votes
1answer
51 views

Finding rotate matrix which solves equation

I try to solve the following problem: given a unit vector v, find rotate matrix R such that R*v = (0,0,..0,1) (vector that it's (n-1) components are 0 and the n'th component is 1). I know that if I ...
0
votes
1answer
44 views

Finding an eigenvectors and eigenvalues to a matrix

I got a question : Given A a matrix which the sum of all elements in each row equals to a constant $\alpha$, find eigenvector and eigenvalue it is belong to. I have no clue from where to start, ...
0
votes
1answer
76 views

If an upper bidiagonal matrix has a repeated singular value, it must have a zero on its diagonal or superdiagonal

I have a question that mentioned in the book "Matrix Computations" by Golub and van Loan. "Show that if $B\in \mathbb{R}^{n\times n}$ is an upper bidiagonal matrix having a repeated singular value, ...
4
votes
1answer
70 views

Is $\left(\sum_{n=0}^\infty\frac{1}{n!}A^n\right)v=\sum_{n=0}^\infty\frac{1}{n!}(A^nv)$?

Suppose we have a convergent power series of matrices $$A=\sum_{n=0}^\infty a_nX^n,$$ for $X\in M_n(\mathbb{C})$. Is it true that if $v\in\mathbb{R}^n$ then $$Av=\sum_{n=0}^\infty a_n(X^nv)?$$ ...
4
votes
3answers
87 views

Are there real solutions to $\exp(X)=-I$?

As we know, the equation $$e^x=-1,\quad x\in\mathbb{C}$$ has no real solution (in fact $x=i\pi+2ki\pi$, $k\in\mathbb{Z}$). I am now considering the generalization of this question to $2\times 2$ ...
1
vote
0answers
235 views

How to find the number of connected components of a graph by using its 16x16 adjacency matrix?

Good day, I have this exercice that provides me with the 16x16 matrix of adjacency of a graph and it asks me to find the number of connected components of the graph and I need to give a spanning tree ...
17
votes
6answers
7k views

Sylvester rank inequality

If $A$ and $B$ are two matrices of the same order $n$, then $$ \operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n. $$ I don't know how to start proving this inequality. I ...
8
votes
7answers
4k views

Why is the inverse of a sum of matrices not the sum of their inverses?

Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$? I know the answer is no, but don't get why.
1
vote
1answer
51 views

Homogenous System

This question caught me off guard. I believe I did it the right way, but it was a bit confusing and I wanted a bit more explanation of the process and to see if what I got was accurate. This is the ...
0
votes
4answers
45 views

If two invertible matrices agree on a vector, does this imply their determinant agrees as well?

As stated, if we let $A, B \in M_n(\mathbb{R})$ be invertible and there is some $v\in R^n$ such that $$Av = Bv$$ does it follow that $\det(A) = \det(B)$? Additionally, does this hold if we let $A, B ...
0
votes
3answers
171 views

Square root of determinant equals determinant of square root?

Is it true that for a real-valued positive definite matrix $X$, $\sqrt{\det(X)} = \det(X^{1/2})$? I know that this is indeed true for the $2 \times 2$ case but I haven't been able to find the answer ...
0
votes
1answer
51 views

Find the inverse of defined operation $\Delta$

We defined the operation $ \Delta $ as $ (a,b) \Delta (c,d) = (ac + \delta bd, ad + bc) $ where $ a, b, c ,d \in \mathbb{Q} $ I have already proven that this operation is both commutative and ...
0
votes
1answer
66 views

Matrix gauss-jordan / gaussian

I am a bit confused in terms of doing gaussian and gauss-jordan elimination for a system of equations. For example let's say we have the following system of equations: We get the following: ...
0
votes
1answer
286 views

How to solve matrix equations involving non invertible matrices?

I need to solve an equation of the form $Ax=b$. Since $A$ is a singular matrix so it doesn't have an inverse, is there a way I can calculate $x$ by solving this equation?
0
votes
1answer
90 views

Limit of regular symmetric matrix

I've got this statement about the topic. I'm trying to figure it out as it is given without proof. I know a symmetric matrix is a square matrix $A$ such that $A = A^T$ and a regular matrix is one ...
0
votes
1answer
49 views

Let $x = (11, 2)^T$ . Find both reflection matrices $M$ such that $Mx$ is a multiple of $e_1$.

How would I go about solving this? I believe my professor said that it deals with householder matrices. I feel like I should calculate $v = x + ||x|| e_1$ and then calculate $u = \frac{v}{||v||}$ ...
0
votes
2answers
46 views

Prove that Det(A-E)=0 if and only if AC=C

We have some $n \times n$ matrix $A$ and $n \times 1$ vector C. Let $E$ be the identity matrix. $$Det(A-E)=0 \iff AC=C.$$ Me and a few friends have been trying to prove it, but none of us could. ...
0
votes
2answers
358 views

Why is the rank of a matrix invariant under row operations?

Prove that the rank of a matrix ($m\times n$) doesn't change if we apply row operations. For example if we multiply a row with a nonzero number $k$.
0
votes
2answers
163 views

Rank of vectors

Prove that the rank of a system of vectors from $E^n$ does is not bigger than the dimension of the vectors. For example the vectors $a,b,c$ are from $E^n$ so each of them has $n$ components (the ...
0
votes
2answers
163 views

A real and normal matrix with all eigenvalues complex but some not purely imaginary?

I'm trying to construct a normal matrix $A\in \mathbb R^{n\times n}$ such that all it's eigenvalues are complex but at least one of then also has a positive real part (well, at least two then, since ...
0
votes
2answers
15 views

A question on Matrix limits

Consider $M(n, \mathbb R)$ , the set of all real square matrices of size $n$ , as an NLS with norm $||A||:=\sqrt{\sum_{i=1}^n \sum_{j=1}^n a_{ij}^2}=Trace(AA^t)$ , then is it true that a sequence of ...
2
votes
1answer
105 views

If $A$ is a square matrix that is linearly independent, is $AA$?

I'm just not sure how to start this problem from Linear Algebra Done Wrong. The problem is to prove that if the columns of $A$, square matrix, are linearly independent, then the columns of $A^2$ = $AA$...
0
votes
1answer
35 views

Transition functions of sheaf tensor product

Suppose $\mathcal{L},\mathcal{M}$ are invertible sheafs on a scheme $X$. I've seen an abstract construction of $\mathcal{L}\otimes_X \mathcal{M}$, but I'm having trouble connecting this with a more ...
2
votes
2answers
156 views

Why use homogeneous coordinates?

I am having trouble understand the use of homogeneous coordinates for when describing transformations in 3D space. From what I have seen, the only difference between a transformation matrix in ...
0
votes
1answer
74 views

Transformations between coordinate systems

I have three three-dimensional orthogonal coordinate systems, O, A and B. A and B are the result of two different transformations from O. I now want to calculate the transformation matrix R, which ...
1
vote
2answers
88 views

Expression for arbitrary powers of a particular $2\times2$ matrix

Given$$\mathbf{M}= \begin{pmatrix} 7 & 5 \\ -5 & 7 \\ \end{pmatrix} $$, what's the formula matrix for $\mathbf{M}^n$? The eigenvalues and eigenvectors are ...
2
votes
1answer
139 views

Simulate correlated $\chi^2$ distribution

I understand that when one have multiple independent variable that follows $N\sim(0,1)$, denoted as $A$ if we have a correlation matrix $R$, we can generate correlated variables $B$ that are normally ...
0
votes
1answer
20 views

eigenvalue of a specific matrix

I am looking for a way to calculate the eigenvalues of this matrix. the last row contains complex numbers in general.
0
votes
1answer
41 views

Finding the value for the reproduction rate that will cause population stabilization

So essentially I have a matrix equation AB A is a 4x4 matrix containing reproduction rate, survival rate and maturity rate. B is a 4x1 matrix containing the populations for each age group. How would ...
0
votes
2answers
37 views
0
votes
0answers
69 views

Decomposition of a matrix into a product of two-level unitaries

I am working through example 4.12 and 4.13 from 'Quantum Computing: From Linear Algebra to Physical Realizations' (Pages 83-84 and 405-406) and I can't seem to figure out some what the '*' in figure (...
2
votes
2answers
105 views

System of Equations

How would I solve a $4 \times 3$ matrix? I've tried making it into an augmented matrix but I ended up with all zeros at the bottom. Please help! $$\begin{align}\begin{cases}x_1+x_2+x_3+x_4&=1 \\...
0
votes
1answer
49 views

Use given identity to computer exponent of 4x4 matrix

I've been given an identity (that I don't know how to prove unfortunately), and been asked to use it to compute exp$(xM)$, where $$ M = \begin{bmatrix} 1 & 1 & 1 & 1 \\ ...
1
vote
1answer
42 views

Finding eigenvalues of a matrix with two unknowns

I've been asked to find the eigenvalues of the following matrix: $$ \begin{bmatrix} 0&1&1\\ 0&0&1\\ 216k^3&-108k^2&18k \end{bmatrix} $$ I'm just not sure how to work it out as ...
0
votes
2answers
66 views

Calculate $e^{xA} $

$$ A = \begin{bmatrix} 1 & 1 & -1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix} $$ I have the answer, but I don't know the ...
1
vote
0answers
53 views

Given a Positive Definite Matrix, find conditions of elements inside the matrix

I have a question that asks me to use the following symmetric positive definite matrix of order $n + 1$ $$B = \begin{bmatrix} \alpha & a^T \\ a & A \end{bmatrix} $$ With this matrix, I ...
1
vote
1answer
32 views

About a determinant identity.

If $A$ is any matrix and $B$ is a rank $2$ matrix of the same dimension then it follows that for any real $t$, $det(A -B) = [1-\partial_p + \frac{1}{2}\partial_p^2 ]det(A + pB) \vert _{p=0}$ I ...
0
votes
0answers
57 views

Given justification for induced norm for nonsingular matrix

Help!! Determine whether the following statement is true or false and give a justifcation: If A is nonsingular, then for any induced norm, $$ ||A^{-1}|| = ||A||^{-1}$$ Thanks!