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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10
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2answers
310 views

A matrix w/integer eigenvalues and trigonometric identity

Any intuition and/or rigorous arguments on the proofs of the following statements would be appreciated: Let $n$ be a natural number. (a) Consider the following Toeplitz/circulant symmetric matrix: ...
10
votes
2answers
165 views

Rank of the difference of matrices [duplicate]

Let $A$ and $B$ be to $n \times n$ matrices. My question is: Is $\operatorname{rank}(A-B) \geq \operatorname{rank}(A) - \operatorname{rank}(B)$ true in general? Or maybe under certain assumptions?
10
votes
3answers
233 views

Prove the inequality: $\prod_{j=1}^ka_{jj}\leq\left(\frac{1}{k}\sum_{j=1}^k\lambda_j\right)^k.$

To all those who are eagerly awaiting a new question, all those who love math, I give this challenge and I hope for you good moments of reflection. Let $A=(a_{ij})_n$ a real nonnegative symmetric ...
10
votes
1answer
134 views

Prove that if $f$ is continuous at $0$, it is continuous on $\mathbb{R}$

Long story short, the question I'm stuck on is as follows: Let $f$ be a positive-definite function. Prove that if $f$ is continuous at $0$, then it is continuous everywhere. Here's the long ...
10
votes
2answers
316 views

Geometric interpretation of normal and anti-hermitian matrices

How do I interpret following types of matrices as special types of transformations? I mean what are the transformative properties of following types of matrices, from $\mathbb{R}^n $ to $ ...
10
votes
1answer
171 views

how prove the following statment for this matrix.

Let $A:=[a_{ij}]_{n×n}$ , $a_{ij}=0$ or $a_{ij}=1$ and $\exists m \in\mathbb N$ such that $A^m=J-I$, where $I$ is the identity matrix and $J=[1]_{n×n}$ (each entry is $1$). How to prove: $\exists ...
10
votes
1answer
161 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
10
votes
1answer
211 views

From matrices to bipartite graphs

Assume $G(A,B)$ is a bipartite graph and assume $L(G)$ is the adjacency matrix of its line graph. define $$B=[3\text{I}+L(G)]^{-1}$$. Is it always the case that for each edge $e=(a,b)\in G$, we have: ...
10
votes
1answer
256 views

Matrix factorization

I'd like to factorize matrices as follows: $$ \left(\begin{array}{cc}X_1&X_2\\X_3&X_4\end{array}\right) = ...
10
votes
1answer
178 views

$\mathcal{O}(n,\mathbb R)$ spans $\mathcal{M}(n,\mathbb R)$

Let $n\geq 3$. One can show that the orthogonal group of degree $n$ over the real field, $\mathcal{O}(n,\mathbb R)$, spans the entire vector space of real $n\times n$ matrices, $\mathcal{M}(n,\mathbb ...
10
votes
0answers
219 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= \begin{bmatrix} 1 ...
10
votes
0answers
335 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
9
votes
4answers
3k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
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votes
7answers
6k views

practical uses of matrix multiplication

The use of matrix multiplication is usually given with graphics initially (scalings, translations, rotations, etc). Then there are more in-depth examples such as counting the number of walks between ...
9
votes
3answers
776 views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ ...
9
votes
3answers
2k views

Rank of skew-symmetric matrix

Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all ...
9
votes
1answer
2k views

Is the matrix $A$ diagonalizable if $A^2=I$

If $A$ is an involutory matrix, i.e. $A^2=I$, then is $A$ diagonalizable?
9
votes
4answers
1k views

What is the fastest way to find the characteristic polynomial of a matrix?

Finding the characteristic polynomial of a matrix of order $n$ is a tedious and boring task for $n > 2$. I know that: the coefficient of $\lambda^n$ is $(-1)^n$, the coefficient of ...
9
votes
4answers
316 views

Example that the Jordan canonical form is not “robust.”

I'm working on this problem that asks to show that the Jordan canonical form is not robust in the sense that small changes in the entries of a matrix $A$ can cause large changes in the entries of its ...
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
3answers
1k views

What's the meaning of the transpose?

I don't understand the motivation of the transpose (or better yet, I haven't even seen one). It feels like just something pulled out of a hat. Thinking about it makes it seem like a product of being ...
9
votes
1answer
841 views

How to understand spectral decomposition geometrically

Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have $$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$ and $$A^{-1} = ...
9
votes
2answers
694 views

Algebraic proof of a trig matrix identity?

I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question: I have a geometric proof, but is there an elegant algebraic proof ...
9
votes
2answers
924 views

$AB-BA=I$ having no solutions

The question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I tried block ...
9
votes
2answers
209 views

Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?

Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
9
votes
2answers
592 views

Has there been a rigorous analysis of Strassen's algorithm?

According to Wikipedia, Strassen's Algorithm runs in $O(N^{2.807})$ time. Has anyone seen a more rigorous analysis displaying constants, possibly in a specific language such as C or Java? I ...
9
votes
1answer
774 views

Matrix raised to a matrix

Good evening, I was wondering if there is such a valid operation as raising a matrix to the power of a matrix, e.g. vaguely, if $M$ is a matrix, is $$ M^M $$ valid, or is there at least something ...
9
votes
1answer
201 views

Calculating the eigenvalues of a matrix

How to find the eigenvalues of $$\begin{bmatrix} 0 & 1 & & &\\ k & 0 & 2 & &\\ & k-1 & 0 & 3 &\\ & ...
9
votes
5answers
2k 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 ...
9
votes
1answer
401 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
9
votes
3answers
311 views

Generation of unimodular matrices with bounded elements

Does anybody know what is the algorithm for generating random unimodular matrices (integer matrices with determinant $\pm 1$) whose elements do not exceed a given bound? Such an algorithm is mentioned ...
9
votes
3answers
399 views

What are mandatory conditions for a family of matrices to commute?

Suppose that there are some matrices. Each matrix in the set must commute with another in the set. What are the mandatory conditions for this?
9
votes
1answer
190 views

Matrix algorithm convergence

Suppose I start with a $n \times n$ matrix of zeros and ones: $$ \begin{bmatrix} 0 & 0 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 ...
9
votes
2answers
396 views

Determine the winner of a tic tac toe board with a single matrix expression?

Assume a tic-tac-toe board's state is stored in a matrix. $$ S=\begin{bmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ \end{bmatrix} $$ Here, $X$ is mapped to $1$, $O$ is ...
9
votes
1answer
258 views

What can we say about $(I-AD)^{-1}$ if $D$ is a diagonal matrix?

Assume we know that square matrices $A$ and $(I-AD)^{-1}$ are invertible and also $D$ is a diagonal matrix. Also assume that $A$ is a symmetric matrix. My question is when we can express ...
9
votes
5answers
407 views

Smallest Non-negative number in a matrix

There is a question I encountered which said to fill an $N \times N$ matrix such that each entry in the matrix is the smallest non-negative number which does not appear either above the entry or to ...
9
votes
1answer
1k views

Does equality of characteristic polynomials guarantee equivalence of matrices?

I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and ...
9
votes
3answers
482 views

Is there a general formula for the derivative of $\exp(A(x))$ when $A(x)$ is a matrix?

It's easy for scalars, $(\exp(a(x)))' = a' e^a$. But can anything be said about matrices? Do $A(x)$ and $A'(x)$ commute such that $(\exp(A(x)))' = A' e^A = e^A A'$ or is this only a special case?
9
votes
1answer
315 views

Direct proof that nilpotent matrix has zero trace

Does anyone knows a proof from the first principles that nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices.
9
votes
1answer
653 views

The Determinant of a Sum of Matrices

Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$, \begin{align} \det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma}, \end{align} ...
9
votes
1answer
709 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 ...
9
votes
2answers
794 views

Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
9
votes
1answer
463 views

Detecting symmetric matrices of the form (low-rank + diagonal matrix)

Let $\Sigma$ be a symmetric positive definite matrix of dimensions $n \times n$. Is there a numerically robust way of checking whether it can be decomposed as $\Sigma = \mathcal{D} + v^t.v$ where $v$ ...
8
votes
6answers
2k views

If $A^2$ is invertible, then $A$ is also invertible?

True or False: If $A^2$ is invertible, then $A$ is also invertible. ($A$ is a matrix here.) The answer is true. I was trying to come up with an example that makes this false. But I couldn't. ...
8
votes
6answers
489 views

$\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)$ not diagonalizable

I would like to ask you about this problem, that I encountered: Show that there exists no matrix T such that $$T^{-1}\cdot \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} ...
8
votes
3answers
551 views

Trace of powers of a nilpotent matrix

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be 0, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like to ...
8
votes
3answers
286 views

$A \in M_{n}(\mathbb{R})$ satisfies $A+A^{t}=I$, then does it imply $\text{det}(A)>0$

I am stuck on this simple question for a long time. If $A \in M_{n}(\mathbb{R})$ satisfies $A+A^{t}=I$, then does it imply $\text{det}(A)>0$? I tried finding a counter-example as well as tried ...
8
votes
4answers
521 views

“weird” ring with 4 elements - how does it arise?

For no real reason, I did a classification of (associative, with a 1) rings with less than 8 elements (they all happen to be commutative). Most of the rings I got were of a type I knew - namely: ...
8
votes
3answers
279 views

Is $\;\det(A^n) =\left(\det (A)\right)^n\;$?

How can the value of $\;\det\left(A^{11}\right)\;$ be calculated from $\;\det(A)$? Generally how can $\;\det\left(A^n\right)\;$ be obtained from $\;\det(A)$?
8
votes
7answers
418 views

Is the square root of a triangular matrix necessarily triangular?

$X^2 = L$, with $L$ lower triangular, but $X$ is not lower triangular. Is it possible? I know that a lower triangular matrix $L$ (not a diagonal matrix for this question), $$L_{nm} \cases{=0 & ...