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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16
votes
5answers
990 views

What's the point of orthogonal diagonalisation?

I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is. The definition is basically this: "A matrix A is ...
16
votes
5answers
18k views

When is matrix multiplication commutative?

I know that matrix multiplication in general is not commutative. So, in general: $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$ But for some matrices, this equations holds, e.g. A = ...
16
votes
4answers
6k views

Determining whether a symmetric matrix is positive-definite (algorithm)

I'm trying to create a program, that will decompose a matrix using the Cholesky decomposition. The decomposition itself isn't a difficult algorithm, but a matrix, to be eligible for Cholesky ...
16
votes
4answers
431 views

Powers of random matrices

Let $M$ be an $n \times n$ matrix whose elements are random reals in [0,1]. Two questions. What is the growth rate of the magnitude of the elements of $M^k$ as a function of $k$? It is definitely ...
16
votes
3answers
959 views

Ratio of largest eigenvalue to sum of eigenvalues — where to read about it?

Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio $$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$ is a measure of the "rank-one-ness" ...
15
votes
4answers
3k views

Intuitive explanation of a positive-semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is ...
15
votes
5answers
629 views

Is this matrix obviously positive definite?

Consider the matrix $A$ whose elements are $A_{ij} = a^{|i-j|}$ for $-1<a<1$ and $i,j=1,\dots,n$ e.g. for $n=4$ the matrix is $$A = \left[ \begin{matrix} 1 & a & a^2 & a^3 \\ a ...
15
votes
5answers
673 views

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
15
votes
4answers
847 views

A matrix is diagonalizable, so what?

I mean, you can say it's similar to a diagonal matrix, it has n independent eigenvectors, etc., but what's the big deal of having diagonalizability? Can I solidly perceive the differences between two ...
15
votes
9answers
883 views

Why do determinants have their particular form?

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant ...
15
votes
4answers
613 views

Determinant of a generalized Pascal matrix

Let $M$ denote the infinite matrix defined recursively by $$ M_{ij} = \begin{cases} 1, & \text{if } i=1 \text{ and } j=1; \\ aM_{i-1,j}+bM_{i,j-1}+cM_{i-1,j-1}, & ...
15
votes
3answers
776 views

Matrices - Conditions for $AB+BA=0$

The Problem Let $A$ be the matrix $\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix} \bigr)$, where no one of $a,b,c,d$ is $0$. Let $B$ be a $2\times 2$ matrix such that ...
15
votes
1answer
131 views
+250

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
15
votes
1answer
151 views

How to calculate the number of factorizations of a square matrix?

I need to write a function, that, given a square matrix M of non-negative integers, calculates the number of representations of M as a product of two square matrices of non-negative integers. Could ...
15
votes
1answer
457 views

Trace inequality for real matrices

Is there any general result characterizing real matrices $A$ such that $$[\mathrm{tr}(A)]^2\leq n\mathrm{tr}(A^2)?$$ I can see that the inequality holds if: all eigenvalues of $A$ are real (by the ...
14
votes
5answers
724 views

Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$?

Is it possible for a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$, $$\mathbf{A}^2=-\mathbf{I}$$ I know that It is possible for $2\times2$ matrix, but is it possible for $3\times3$ matrix ?
14
votes
5answers
723 views

A symmetric matrix whose square is zero

I was once asked in an oral exam whether there can be a symmetric non zero matrix whose square is zero. After some thought I replied that there couldn't be because the minimal polynomial of such a ...
14
votes
4answers
559 views

Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$

I was reviewing some matrices and found this interesting if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
14
votes
5answers
6k views

Differences between a matrix and a tensor

What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
14
votes
5answers
2k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
14
votes
3answers
962 views

Why is the determinant of a symplectic matrix 1?

suppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix} 0 & E_n\\ -E_n&0 \end{pmatrix}$$ where $E_n$ represents identity matrix. if $A$ satisfies $$AJA^T=J$$ How to figure out ...
14
votes
2answers
757 views

Why is the trace of a matrix the sum along its diagonal?

Define the trace of a matrix with entries in $\mathbb C$ to be the sum of its eigenvalues, counted with multiplicity. It is a standard (but I think extremely surprising) fact that this is the sum of ...
14
votes
2answers
369 views

easier way of calculating the determinant for this matrix

I have to calculate the determinant of this matrix: $$ \begin{pmatrix} a&b&c&d\\b&c&d&a\\c&d&a&b\\d&a&b&c \end{pmatrix} $$ Is there an easier way of ...
14
votes
4answers
978 views

How does multiplying by trigonometric functions in a matrix transform the matrix?

I found this comic: But I can't understand the humor because I can't understand how trig functions affect matrix multiplication. Can someone please explain?
14
votes
3answers
7k views

Physical Meaning of Null Space of a Matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions... my textbook gives me the definition, but I just don't "get" it. E.g.: I ...
14
votes
1answer
501 views

Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $ A_{ij}=\frac{1}{i + j - 1}$. I need to show that $A$ is invertible and the inverse has integer entries. I was able to show that $A$ is invertible. How do I ...
14
votes
2answers
230 views

Why does calculating matrix inverses, roots, etc. using the spectrum of a matrix work?

Suppose $A$ is a $n \times n$ matrix from $M_n(\mathbb{C})$ with eigenvalues $\lambda_1, \ldots, \lambda_s$. Let $$m(\lambda) = (\lambda - \lambda_1)^{m_1} \ldots (\lambda - \lambda_s)^{m_s}$$ be the ...
14
votes
1answer
225 views

What do characteristic polynomials characterize?

Let $R$ be an integral domain and $F$ a finitely generated free module over $R$. For a linear transformation $\alpha\in\operatorname{End}_R(F)$, the characteristic polynomial is \begin{equation} ...
13
votes
7answers
972 views

How can I show that $\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$?

Well, the original task was to figure out what the following expression evaluates to for any $n$. $$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^{\large n}$$ By trying out different values of ...
13
votes
8answers
3k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let A and B be two matrices which can be multiplied. Then $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$ I proved $\operatorname{rank}(AB) \leq ...
13
votes
4answers
857 views

Is $A + A^{-1}$ always invertible?

Let $A$ be an invertible matrix. Then is $A + A^{-1}$ invertible for any $A$? I have a hunch that it's false, but can't really find a way to prove it. If you give a counterexample, could you please ...
13
votes
10answers
421 views

$2\times2$ matrices are not big enough

Olga Tausky-Todd had once said that "If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." There are, however, assertions about matrices that are true for ...
13
votes
4answers
4k views

Show that the determinant of $A$ is equal to the product of its eigenvalues.

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
13
votes
4answers
474 views

Expected Value of a Determinant

Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$. My conjecture is ...
13
votes
4answers
369 views

Choosing an orthonormal basis in which a linear operator has a sparse matrix

Given a linear operator $T$ on an $n$-dimensional vector space $V$ (over $\mathbb R$), I want to find an orthonormal basis for $V$ in which the matrix of $T$ is sparse (has many zeros). How many zeros ...
13
votes
3answers
396 views

If $C$ commutes with certain matrices $A$ and $B$, why is $C$ a scalar multiple of the identity?

I'm self studying Steven Roman's Advanced Linear Algebra, and this is problem 10 of Chapter 8. Let $A,B\in M_2(\mathbb{C})$, $A^2=B^3=I$, $ABA=B^{-1}$, but $A\neq I$ and $B\neq I$. If $C\in ...
13
votes
3answers
811 views

How to show determinant of a specific matrix is nonnegative

How to show that $$\det A= \det \begin{pmatrix}\cos\frac{\pi}{n}&-\frac{\cos\theta_1}{2}&0&0&\cdots&0&-\frac{\cos\theta_n}{2} ...
13
votes
2answers
2k views

Topology of matrices

1.Consider the set of all $n×n$ matrices with real entries as the space $\mathbb R^{n^2}$ . Which of the following sets are compact? (a) The set of all orthogonal matrices. (b) The set of all ...
13
votes
1answer
808 views

Max dimension of a subspace of singular $n\times n$ matrices

I am sure the answer to this is (kind of) well known. I've searched the web and the site for a proof and found nothing, and if this is a duplicate, I'm sorry. The following question was given in a ...
13
votes
3answers
355 views

when does $\det(AB^T+BA^T)\le \det(AA^T+BB^T)$ hold?

When does the following matrix inequality hold? $$\det(AB+B^TA^T)\le \det(AA^T+BB^T)$$ $A$ and $B$ are any real matrices. My reply gives a counter example. The question is under what condition ...
12
votes
8answers
1k views

I need to calculate $x^{50}$ [duplicate]

$x=\begin{pmatrix}1&0&0\\1&0&1\\0&1&0\end{pmatrix}$, I need to calculate $x^{50}$ Could anyone tell me how to proceed? Thank you.
12
votes
2answers
658 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
12
votes
5answers
314 views

Matrices with $A^3+B^3=C^3$

Problem: Find infinitely many triples of nonzero $3\times 3$ matrices $(A,B,C)$ over the nonnegative integers with $$A^3+B^3=C^3.$$ My proposed solution is in the answers.
12
votes
3answers
1k views

Why do the $n \times n$ non-singular matrices form an “open” set?

Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context. Thank you.
12
votes
5answers
272 views

Prove that $A^k = 0 $ iff $A^2 = 0$

Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$. I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this ...
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.
12
votes
5answers
7k views

Matrix Inverses and Eigenvalues

I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were ...
12
votes
4answers
266 views

Is it true that all matrices in $M_2(\mathbb R)$ is the sum of two squares?

I recently show that every polynomial with real coefficient and $P$ is always positive. is a sum of two squares of polynomials. These questions also appear often in arithmetic. What if we change ...
12
votes
4answers
6k views

What is the geometric meaning of singular matrix

Could anyone help explain what is the geometric meaning of singular matrix ? What's the difference between singular and non-singular matrix ? I know the definition, but couldn't understand it very ...
12
votes
3answers
184 views

$\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$ for every positive definite matrix $A$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}e^{-\langle Ax,x\rangle}\text{d}x=\left|\,\det\left(\pi^{-1}{A}\right)\right|^{-1/2}\!, $$ where ...