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), ...

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-1
votes
2answers
69 views

Prove $f(A^T)=f(A)^T$ for a matrix $A$ [closed]

As the title says, I need to prove $f(A^T)=f(A)^T$ for a matrix $A$. (where $T$ is the transpose) I believe the proof involved the fact that an interpolation polynomial $r(A)=f(A)$ and then I must ...
5
votes
1answer
111 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
0
votes
3answers
1k views

Why is only a square matrix invertible?

Can anyone give a very simple proof (or explanation) as to why only square matrix can possibly be invertible?
0
votes
0answers
106 views

Nilpotency of the adjacency matrix of a directed tree network

Say I have a directed network that is organized in a tree, with all connections going downstream (genealogically). By that I mean that there is one root node connected to $c_{00}$ child nodes, and ...
1
vote
1answer
26 views

Finding the values of a vector if the vector.matrix product and the value of the matrix is known (only using left multiplication operations)

Given an unknown input vector V (v1, v2, v3, v4), a known matrix A [a1... a4; b1..b4;..etc.] and a known vector.matrix product M [m1....m4]. Can you discover V? Normally you would just take the ...
6
votes
1answer
67 views

Is $\mathrm{GL}_n(K)$ divisible for an algebraically closed field $K?$

This is a follow-up question to this one. To reiterate the definition, a group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ ...
5
votes
2answers
83 views

Is $\mathrm{GL}_n(\mathbb C)$ divisible?

A group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Is the group $\mathrm{GL}_n(\mathbb C)$ divisible? Or more precisely, ...
-3
votes
1answer
79 views

How do I multiply a $2 \times 2$ matrix by $i$ in a summation formula?

How do I multiply a $2 \times 2$ matrix by the power of $i$ in a summation formula? $$\left[ \begin {array}{cc} a&b\\ c&d\end {array} \right] ^{i}$$ I want to get the values of the first ...
0
votes
1answer
48 views

Can this function with modulo and truncated division be simplifed?

Can this function with modulo and truncated division (DIV) be simplifed? f(x)=(x%c)*r+DIV(x,c)%r Basically, I use this ...
0
votes
2answers
94 views

How to transform between two layout forms of matrix calculus?

I'm trying to derive a very simple matrix derivative : take derivative of Tr(A' X) with respect to X. However, I got two different answers by following different methods. First Method: vec routine: ...
0
votes
1answer
256 views

Null space of this matrix is no solution?

When I set this row reduced matrix (which I row reduced using matlab) equal to zero for finding the null space, am I supposed to get no solution? Because x8, the last vector will be equal to zero.
0
votes
0answers
75 views

Calculus with Exponential Matrix

I have a following with derivating and integrating using exponential Matrix. Kindly have a look at it. Consider $A\in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}$, $u(t)\in ...
1
vote
1answer
90 views

Eigenvalues of discretized linear integral operator

Suppose I have the following kernel operator: $Af(x) = \int_{-1}^1 K(x-y)f(y)dy$ which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
1
vote
2answers
249 views

Multiple Choice question about an $n \times n$ matrix $A$ with real or complex entries, and such that $A^3=0$

Let $A$ be an $n \times n$ matrix with real or complex entries and such that $A^3=0.$ Which of the following options holds? 1. $(I+A)^3=0$. 2. $I+A$ is invertible. 3. $I+A$ is not invertible. ...
-1
votes
1answer
145 views

Confusion related to derivative of a quadratic equation

This might be a simple question but how come F(x) = x'Ax F'(x) = (A + A')x I didn't get it
0
votes
0answers
112 views

What is the name for a non-square permutation matrix?

Consider a matrix that selects and permutes some but not all of the entries of a vector. That is a binary $n\times m$ matrix, where $n<m$, with a single one per row, for example ...
1
vote
0answers
157 views

Power Series and Matrices

I am trying to prove that if a function $f(x)$ can be written as a power series in the form \begin{equation} f(x)=\sum_{n=0}^{\infty}c_n(x-x_0)^n \end{equation} such that $|x-x_0|<r$, then ...
5
votes
3answers
544 views

Describe all matrices similar to a certain matrix.

Math people: I assigned this problem as homework to my students (from Strang's "Linear Algebra and its Applications", 4th edition): Describe in words all matrices that are similar to ...
1
vote
0answers
45 views

root of binary matrix

There is a square matrix A defined over the field GF(2). It means there are zeros and ones in its cells, xor stands for element summation, logical and - for multiplication. Is there any way to find ...
2
votes
2answers
176 views

Link between the norm $1$ of a matrix and its biggest eigenvalue

I am working on a set of matrices for a project, studying their highest eigenvalue, let's call it $\lambda_{1}$. I was curious and plotted the norm 1 of the matrix, ie $ \frac{1}{n^{2}}\sum_{i,j} ...
2
votes
0answers
178 views

the eigenvalues of product of matrices

Let $A$ be an invertible matrix with positive eigenvalues and $B$ be a positive definite matrix . How to estimate the minimum eigenvalues of $AB$ by using the eigenvalues of $A$ and $B$. More ...
2
votes
0answers
44 views

Are decomposable maps completely bounded?

By the word decomposable I mean a positive map $\phi:\mathcal{B(H)}\rightarrow \mathcal{B(K)}$; $\mathcal{H,K~}$ are some Hilbert spaces and $\phi=\psi_1+T\circ \psi_2$ where $T$ is the transpose ...
2
votes
1answer
98 views

Prove the mapping $(x,y,z)\mapsto (x+e^y,y+e^z,z+e^x)$ is locally invertible.

Show that the mapping $\mathbb{R}^3\to \mathbb{R}^3$, $(x,y,z)\mapsto (u,v,w)$ which is defined by $$\begin{align*} u&=x+e^y\\ v&= y+e^z\\ w &=z+e^x \end{align*}$$is locally invertible ...
4
votes
1answer
58 views

Matrix decomposition into two arbitrary sized matrices

Given a matrix $A$ of dimensions $m\times{}n$, I am interested in decomposing $A$ into the product $BC$ where $B$ is a $m\times{}p$ matrix and $C$ is a $p\times{}n$ matrix. What are the methods to ...
0
votes
1answer
38 views

Square root entries of matrices

How would you simplify something like this? $((\xi'\omega \xi)^{-1})^{0.5}$ where $\xi$ is a $k \times 1$ matrix, $\omega$ is a $k\times k$ square matrix. Thank you very much! Edit: Yes, though ...
1
vote
1answer
246 views

Is M (a non-symmetric matrix) positive definite if the product NM is positive definite where N is a diagonal positive definite matrix.

If the product of two matrices, N (a diagonal positive definite matrix) and M (a non-symmetric matrix), is positive definite i.e. $x^TNMx>0$, then is the matrix M positive definite i.e. is ...
1
vote
1answer
379 views

Project Euler $420$ [closed]

So the question is: We define $F(N)$ as the number of the $2\times 2$ positive integer matrices which have a trace less than $N$ and which can be expressed as a square of a positive integer matrix ...
3
votes
0answers
73 views

How to solve a distance problem inside of a picture?

sorry for my bad english. I have the following problem: In the picture you can see 4 different positions. Every position is known to me (longitude, latitude with screen-x and screen-y). Now i want ...
0
votes
0answers
155 views

Is this subset of positive definite symmetric matrices closed?

Consider the collection of all matrices of the form $$\Lambda \Lambda^T + \Psi$$ where $\Psi$ is $n \times n$,positive definite, and diagonal and $\Lambda$ is $n \times m$ with $m < n$. In ...
2
votes
2answers
704 views

How do I write this matrix in Jordan-Normal Form

I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors ...
1
vote
4answers
408 views

Eigenvalues of a matrix with only one non-zero row and non-zero column.

Here is the full question. Only the last row and the last column can contain non-zero entries. The matrix entries can take values only from $\{0,1\}$. It is a kind of binary matrix. I am ...
2
votes
1answer
168 views

Topologically equivalence of a metric on matrices

Define a function on the set of $n\times n$ matrices by $\rho(A,B)=\operatorname{rank}(A-B).$ Prove that $\rho$ is a metric that is topologically equivalent to the discrete metric.
2
votes
1answer
69 views

Disable one angle of rotation

I'd like to disable one angle of rotation of an object rotating in 3D space. Imagine a camera rotating around and displaying objects as they are in space. I'd like this object to be fixed on the ...
2
votes
1answer
62 views

If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.

Let $H$ be an $n\times n$ matrix with entries $\pm1$. Its rows are mutually orthogonal. If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.
2
votes
2answers
45 views

why this is correct: $\det(C+Di)$ is not zero, then there exists some real number $a$ such that $\det(C + a D)$ is not zero

I wonder why the following statement is correct: supposing $C$ and $D$ are two real matrix, if the determinant of the complex matrix $C + D i $ is not zero, then there exists some real number $a$ ...
1
vote
4answers
107 views

If a real matrix induces an isometry, is the matrix orthogonal? Where can I find a proof?

Math people: This is a pretty simple question, but I am having trouble finding an answer. I did not find an answer in the Similar Questions, and I apologize ahead of time if this is a duplicate. ...
3
votes
1answer
112 views

Diagonalizable Matrix $A^2$

How can I find a matrix $A$ such that $A^2$ is diagonalizable but $A$ is not? I have tried many different ways, but to no avail. Is there something that I am missing in the question that gives a ...
2
votes
2answers
40 views

Recovering matrix elements from a matrix equation

I have the following matrix equation: $$\overrightarrow{y}=H\overrightarrow{x}$$ where $H$ is a square matrix of $N$ elements, $y$ is a $N$ columns vector and $x$ a $N$ rows vector. Knowing $x$ and ...
2
votes
1answer
74 views

Proof for certain matrix results?

There are certain results of matrices that Stephen Boyd uses often in his book on Convex optimization. Can someone provide me proof for the results I have enumerated below: If $B \in S^n$ and $A \in ...
10
votes
7answers
629 views

Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?

I saw the following statement in my homework and we are asked to make use of the statement: If $S$ is a symmetric matrix then $$\det(I + S ) = 1 + \operatorname{trace}(S).$$ However, I am not ...
1
vote
0answers
157 views

Criterion for detecting rank-deficiency via QR decomposition?

I apologize in advance if this is an ill-posed question -- I'd appreciate advice on what pieces are missing as much as an answer. I'm solving a system like $P \approx X Y^T$, where P is a large ...
1
vote
1answer
124 views

Matrix completion

I need to find an algorithm (if exists) of the following matrix completion problem. I need to construct $n^2$ positive semi-definite matrices, say $\{P_i\}_{i=1}^n$. Entries of these matrices are ...
1
vote
3answers
151 views

How to prove $BA=0 \implies $ nullity $B \geq $ rank $A.$

I am stuck on the following problem and do not know how to progress: Let $A$ and $B$ be $n \times n$ real matrices. Then how can I prove that $BA=0 \implies $ nullity $B \geq $ rank $A.$ ...
12
votes
3answers
493 views

Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible

I came across the following problem that says: Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible. Then how can I prove the following: rank $A$+ ...
7
votes
3answers
251 views

Matrix Determinant Identity

I have come across an observation about the determinant of a matrix, but I don't know how to prove it in general. Let me demonstrate it through an example. $$ \begin{align} \left| \begin{matrix} 1 ...
2
votes
1answer
111 views

Eigen Values of A?

I just got a quick practice question here that I think should be simple but I can't find a definitive answer. Let $A$ be a square matrix such that $A^3=A$. What can you say about the eigen values of ...
2
votes
1answer
216 views

How can I convert a NxN Matrix to a Vector Nx1?

$$ \left[ \begin{array}{@{}ccccc@{}} 0.9& 0.1& 0& 0& 0& 0& \\ 0& 0.9& 0.1& 0& 0& 0& \\ 0& 0& 0.9& 0& 0& 0.1& \\ 0& 0& ...
0
votes
2answers
233 views
3
votes
2answers
403 views

Differentiate between column space, dimension of column space, and basis of column space.

Say if there is a matrix A: $$\begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ What the column space of A? : I am confused whether ...
0
votes
2answers
164 views

Matrices Problem with 3 Unknown Variables J, K and M

Given: $$ \left[\matrix{1&3 \\-2&4 }\right]+ \left[\matrix{11&5 \\-6&12 }\right]=K\left(\left[\matrix{3&2 \\J&M }\right]\right) $$ Find the value of $J+K+M.$ the answer is $6$ ...