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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2
votes
1answer
993 views

How to prove that exponential kernel is positive definite?

The exponential kernel is defined by: $$k(x,z) = e^{-\alpha\|x-z\|}$$ where $\alpha>0$, $x,z\in \Bbb{R}^d$, $\|x\|$ is the 2-norm. The kernel matrix is defined by $K_{ij} = k(x_i,x_j)$, ...
2
votes
3answers
465 views

Orthogonal and symmetric Matrices

What can one say about the set of all $n$-dimensional square matrices $A \in \text{GL}_n(\mathbb{C})$ that have an inverse with entries out of $\mathbb{C}$ with the properties: unitary ...
4
votes
6answers
664 views

How to find the exact value of $ \cos(36^\circ) $?

The problem reads as follows: Noting that $t=\frac{\pi}{5}$ satisfies $3t=\pi-2t$, find the exact value of $$\cos(36^\circ)$$ it says that you may find useful the following identities: ...
4
votes
2answers
1k views

Power of a matrix

$A$ is a $n\times n$ matrix, $ A^m = 0 $ for some positive integer $m$. Show that $A^n = 0$. My attempt: For $n > m$, it's obvious since matrix multiplication is associative. For $n < ...
2
votes
1answer
387 views

formula for calculating determinant of the block matrix

I saw a formula on the wikipedia page about determinant that $\det\begin{bmatrix}A & B\\ C & D \end{bmatrix}$ = $\det(AD-BC)$, if $C$ and $D$ commute. Is this always true? Or is there a good ...
2
votes
2answers
150 views
2
votes
1answer
296 views

$m\times n$ matrix with an even number of 1s in each row and column

So I want to find the number of ways to fill an $m\times n$ matrix with only 0s and 1s such that each row and column has an even number of 1s. I'm pretty stumped here. I've set up m+n equations ...
1
vote
2answers
195 views

Groups/Linear maps

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...
0
votes
1answer
508 views

what are pivot numbers in LU decomposition? please explain me in an example

studying many pages like wikipedia, wolfram, Mathworks, Math Stack Exchange, lu-pivot, LU_Decomposition I couldn't find exactly whart are the pivot numbers in a specific example: for example what are ...
6
votes
1answer
785 views

If Ax = Bx for all $x \in C^{n}$, then A = B.

Let $A$ and $B$ are nxn matrices and $x \in C^{n}$. If $Ax = Bx$ for all $x$ then $A = B$. To prove this I have selected $x$ from Euclidean basis B = {$e_{1},e_{2},...,e_{n}$}. Then $Ae_{i} = Be_{i}$ ...
4
votes
1answer
254 views

Does the inverse of this matrix of size $n \times n$ approach the zero-matrix in the limit as $\small n \to \infty$?

Fiddling with another (older) question here I constructed an example-matrix of the type $\small M_n: m_{n:r,c} = {1 \over (1+r)^c } \quad \text{ for } r,c=0 \ldots n-1 $ . I considered the inverse ...
4
votes
4answers
4k views

Properties of zero-diagonal symmetric matrices

I'm interested in the properties of zero-diagonal symmetric (or hermitian) matrices, also known as hollow symmetric (or hermitian) matrices. The only thing I can come up with is that it cannot be ...
3
votes
2answers
60 views

Is this fact about matrices and linear systems true?

Let $A$ be a $m$-by-$n$ matrix and $B=A^TA$. If the columns of $A$ are linearly independent, then $Bx=0$ has a unique solution. If is true, can you help me prove it? If is false, could you give a ...
3
votes
2answers
451 views

Why does the $n$-th power of a Jordan matrix involve the binomial coefficient?

I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$-th power is: $$ J_k(\lambda)^n = \begin{bmatrix} \lambda^n & \binom{n}{1}\lambda^{n-1} & ...
3
votes
5answers
270 views

$\left \| \cdot \right \|$ is an induced norm. If $\left \| A \right \|<1$, how to show that $I-A$ is nonsingular and …?

The induced norm $\left \| \cdot \right \|$ is defined for a matrix $A\in\mathbb{C}^{n\times n}$ as $\left \| A \right \|=\sup_{||x||=1} \left \| Ax \right \|$. If $\left \| A \right \|<1$, show ...
3
votes
3answers
1k views

$A^{T}A$ positive definite then A is invertible?

Say if $A$ is an $n \times n$ matrix, why is it that if $A^{T}A$ is positive definite, the matrix $A$ is then invertible? All I know is $A^{T}A$ gives a symmetric matrix but what does $A^{T}A$ is ...
1
vote
4answers
2k views

Square root of Positive Definite Matrix

Let $A$ be an $n\times n$ positive definite matrix. Show that there exists a unique positive definite matrix $B$ such that $B^2=A$. I do know the existence. But what about the uniqueness? Would you ...
1
vote
1answer
1k views

What is a idempotent matrix?

I would like to know what is a idempotent matrix? Also, which invertible matrices are also idempotent and can a matrix be nilpotent and idempotent at the same time?
1
vote
1answer
419 views

Iterating over all matrices with fixed row and column sums

Can anyone suggest an algorithm for iterating once through all matrices with non-negative integer entries which are $2$ by $n$ with fixed row sums ($r_1$ and $r_2$) and fixed column sums ($c_1, c_2, ...
27
votes
2answers
867 views

Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$?

Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any ...
23
votes
7answers
2k views

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
12
votes
4answers
4k views

How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
9
votes
4answers
3k views

Solving very large matrices in “pieces”

Say you have a very dense matrix that is 30000x30000 elements. The very dense matrix comes from the radiosity equation, which I discussed here. Say you have Ax = B. You have B, and A is 30000x30000 ...
17
votes
2answers
1k views

Is it always true that $\det(A^2+B^2)\geq0$?

Let $A$ and $B$ be real square matrices of the same size. Is it true that $$\det(A^2+B^2)\geq0\,?$$ If $AB=BA$ then the answer is positive: ...
16
votes
9answers
3k views

Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &amp;1\\-1&amp;0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
12
votes
3answers
488 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 $A$+ ...
4
votes
5answers
775 views

How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior ...
3
votes
3answers
213 views

Solving inhomogenous ODE

I have an inhomogenous ODE. The main issue here is variables are matrices. It is bit of matrix calculus. A solution would be highly appreciated interms of x . I guess we can use same methods for ...
11
votes
4answers
349 views

How to show path-connectedness of $GL(n,\mathbb{C})$

Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
9
votes
3answers
1k views

$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.

The problem is from Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful. Let $A,B$ be $m \times n$ and $n ...
9
votes
1answer
753 views

Do these matrix rings have non-zero elements that are neither units nor zero divisors?

Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with entries in $R$. In addition, suppose that $R$ is a ring in which every non-zero element is ...
7
votes
1answer
326 views

Holomorphic function of a matrix

A statement is made below. The questions are: (a) Is the statement true? (b) If it is, does it appear in the literature? Here is the statement. For any matrix $A$ in $M_n(\mathbb C)$, write ...
5
votes
2answers
417 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
5
votes
1answer
2k views

Intersection of conics using matrix representation

I came across a very interesting section of a wikipedia article on conics: http://en.wikipedia.org/wiki/Conic_section#Intersecting_two_conics I am trying to work out a couple of examples to add to ...
5
votes
1answer
347 views

Rank of an interesting matrix

Lets define: $U=\left \{ u_j\right \} , 1 \leq j\leq N= 2^{L},$ the set of all different binary sequences of length $L$. $V=\left \{ v_i\right \} , 1 \leq i\leq M=\binom{L}{k}2^{k},$ the set of ...
4
votes
1answer
149 views

If $A^kX=B^kY$ for all $k$, is $X=Y$?

This one is more general than the one I asked before. Given invertible matrices $A,B$ and matrices $X,Y$ all with size $n$, such that $A^k X = B^k Y$ for $k=1,2,...,2n$. Does it follow that $X = Y$? ...
4
votes
1answer
335 views

eigen decomposition of an interesting matrix

Lets define: $U=\left \{ u_j\right \} , 1 \leq j\leq N= 2^{L},$ the set of all different binary sequences of length $L$. $V=\left \{ v_i\right \} , 1 \leq i\leq M=\binom{L}{k}2^{k},$ the set of ...
3
votes
1answer
266 views

Number of $(0,1)$ $m\times n$ matrices with no empty rows or columns

I am looking to calculate the number of $m\times n$ matrices which have no empty rows or columns (at least one $1$ in each row and column). I have looked at the answers to a few similar questions ...
7
votes
2answers
670 views

$AB-BA$ is a nilpotent matrix if it commutes with $A$

I saw this in a MathOverflow post and am putting it here for posterity. Problem: Let $A$ and $B$ by square matrices and set $C=AB-BA$. If $AC=CA$, prove $C$ is nilpotent.
6
votes
3answers
213 views

Prove that $e^{-A} = (e^{A})^{-1}$

Let $A, B \in R^{n \times n}$. Prove that $e^{-A} = (e^{A})^{-1}$. ($R$ is the real numbers) I've tried messing around with both sides, evaluated as sums. I just can't get the two to match up. Any ...
6
votes
1answer
1k views

Block inverse of symmetric matrices

Let us assume we have a symmetric $n \times n$ matrix $A$. We know the inverse of $A$. Let us say that we now add one column and one row to $A$, in a way that the resulting matrix ($B$) is an $(n+1) ...
4
votes
2answers
865 views

Are there any decompositions of a symmetric matrix that allow for the inversion of any submatrix?

I am given a $J \times J$ symmetric matrix $\Sigma$. For a subset of $\{1, ..., J\}$, $v$, let $\Sigma_{v}$ denote the associated square submatrix. I am in need of an efficient scheme for inverting ...
3
votes
1answer
95 views

Find all $3 \times 3$ matrices X, such that $X^2+E=0$

Task is to find all $3 \times 3$ matrices X, $x_{ij} \in R$, such that $X^2+E=0$ I used suggestions from this question, though I stuck anyway. $X^2=Y=-E$ Then $det(Y-\lambda E)=0$, which results ...
2
votes
2answers
303 views

The relationship between eigenvalues of matrices $XY$ and $YX$

If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove that $\lambda (XY) = \lambda (YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$? Here, ...
2
votes
1answer
457 views

How do I prove that in every commuting family there is a common eigenvector?

The proof given by my textbook is highly non-satisfying. The author adopted some magic-like "reductio ad absurdum" and the proof (although is correct) didn't reveal the nature of this problem. I made ...
7
votes
4answers
1k views

eigenvalues of certain block matrices

This question inquired about the determinant of this matrix: $$ \begin{bmatrix} -\lambda &1 &0 &1 &0 &1 \\ 1& -\lambda &1 &0 &1 &0 \\ 0& ...
6
votes
2answers
581 views

is a one-by-one-matrix just a number (scalar)?

I was wondering. Clearly, we cannot multiply a (1x1)-matrix with a (4x3)-matrix; However, we can multiply a scalar with a matrix. This suggests a difference. On the other hand, I was, for example, in ...
4
votes
1answer
177 views

Invertible antisymmetric matrix and identities

A link to the page is available here. The relevant bit is on P. 15 of the book. I would really appreciate it if somebody could help! It is probably something quite obvious, hence left out by the ...
3
votes
1answer
190 views

Matrix, Ranks and Rows

Let $f:V \rightarrow W$ be a linear transformation. Given bases $\{v_i\}_{1\leq i \leq n}$ and $\{w_j\}_{1\leq j \leq m}$ of V and W, respectively, $f$ has an associated $m \times n$ matrix $A$. I am ...
3
votes
1answer
246 views

Linear Algebra,regarding commutator

Suppose $A$ and $B$ are real or complex $n \times n$ matrices and $C = [A,B]$ is their commutator. If $C$ commutes with $A$, show that $C$ is nilpotent.