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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22
votes
1answer
16k views

Simultaneously Diagonalizable Proof

Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a.) Show ...
22
votes
4answers
2k views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
22
votes
4answers
721 views

Can every nonsingular $n\times n$ matrix with real entries be made singular by changing exactly one entry?

I was just thinking about this problem: Can every nonsingular $n\times n$ matrix with real entries be made singular by changing exactly one entry? Thanks for helping me.
22
votes
3answers
818 views

If $\,A^k=0$ and $AB=BA$, then $\,\det(A+B)=\det B$

Assume that the matrices $A,\: B\in \mathbb{R}^{n\times n}$ satisfy $$ A^k=0,\,\, \text{for some $\,k\in \mathbb{Z^+}$}\quad\text{and}\quad AB=BA. $$ Prove that $$\det(A+B)=\det B.$$
22
votes
5answers
19k 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 ...
22
votes
1answer
5k 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} ...
21
votes
6answers
18k views

if eigenvalues are positive, is the matrix positive definite?

If the matrix is positive definite, then all its eigenvalues are strictly positive. Is the converse also true? That is, if the eigenvalues are strictly positive, then matrix is positive definite? Can ...
21
votes
4answers
3k views

Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?

Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all ...
21
votes
3answers
529 views

Multiplying by a $1\times 1$ matrix?

For matrix multiplication to work, you have to multiply an $m \times n$ matrix by an $n \times p$ matrix, so we have $$\bigg(m \times n\bigg)\bigg( n\times p \bigg).$$ But what about a $1 \times 1$ ...
21
votes
5answers
1k views

Is there anything special about this matrix?

I've just encountered a matrix which seems to display nothing special to me: $$B=\begin{pmatrix}1&4&2\\0 &-3 &-2\\ 0 &4 &3 \end{pmatrix}$$ But further observation reveals ...
21
votes
2answers
436 views

Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer

Let $x_1,...,x_n$ be distinct integers. Prove that $$\prod_{i<j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$ I know there is a solution using determinant of a matrix, but I can't remember it now. Any ...
21
votes
1answer
945 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
20
votes
12answers
612 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 ...
20
votes
5answers
2k views

Every Function in a Finite Field is a Polynomial Function

From a bank of past master's exams I am going through: Let $F$ be a finite field. Show that any function from $F$ to $F$ is a polynomial function. I know that finite fields are fields of $p$ ...
20
votes
3answers
737 views

Complexity of computing $ABx$ depends on the order of multiplication

To calculate $y = ABx$ where $A$ and $B$ are two $N\times N$ matrices and $x$ is an vector in $N$-dimensional space. there are two methods. One is to first calculate $z=Bx$, and then $y = Az$. This ...
20
votes
2answers
2k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
20
votes
3answers
684 views

Old AMM problem

I am working on an old AMM problem: Suppose $A,B$ are $n\times n$ real symmetric matrices with $\operatorname{tr} ((A+B)^k)= \operatorname{tr}(A^k) + \operatorname{tr}(B^k) $ for every positive ...
19
votes
8answers
9k views

Do all square matrices have eigenvectors?

I came across a video lecture in which the professor stated that there may or may not be any eigenvectors for a given linear transformation. And, so far I thought every matrix has eigenvectors. Please ...
19
votes
2answers
2k 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: ...
19
votes
3answers
28k views

A matrix and its transpose have the same set of eigenvalues

Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma(A^T)$ where $A^T$ is the transposed matrix of $A$.
19
votes
2answers
391 views

How find this determinant $\det(\cos^4{(i-j)})_{n\times n}$

Question: Define the matrix $A_{k}=(a^k_{ij})_{n\times n}\quad$where $a_{ij}=\cos{(i-j)},\quad n\ge 6$ Find the value $$\det(A_{4})=\:?$$ My try: since $$\det(A_{4})=\begin{vmatrix} ...
18
votes
11answers
481 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? [duplicate]

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
18
votes
8answers
7k 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 $r = \left(\begin{smallmatrix}0 &1\\-1 & 0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
18
votes
6answers
6k views

Matrices which are both unitary and Hermitian

Matrices such as $$ \begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{bmatrix} \text{ or } \begin{bmatrix} \cos\theta & i\sin\theta \\ ...
18
votes
6answers
12k 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 ...
18
votes
5answers
2k 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 ...
18
votes
4answers
2k 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 ...
18
votes
2answers
237 views

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 $q^3+q^2-q$ ...
17
votes
5answers
2k 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 ...
17
votes
4answers
1k views

A problem on Condition $\det(A+B)=\det(A)+\det(B)$

Let $A$ be a matrix $n\times n$ matrix that for any matrix $B$ we have $\det(A+B)=\det(A)+\det(B)$. If this imply that $A=0$? or $\det(A)=0$?
17
votes
5answers
4k 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 ...
17
votes
7answers
6k 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 ...
17
votes
3answers
816 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 $A$+ ...
17
votes
2answers
274 views

Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
17
votes
2answers
5k views

Integral of matrix exponential

Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem \begin{align*} \dot{x}(t) = A x(t), \quad x(0) = x_0 \end{align*} is given by $x(t) = \mathrm{e}^{At} x_0$. I am ...
17
votes
4answers
773 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}, & ...
17
votes
2answers
248 views

A curious property of orthogonal matrices

Let $A$ be an $n\times n$ orthogonal matrix, i.e. $AA^T=A^TA=I$. I noticed experimentally that if we take any increasing set of indices $i_1<\cdots<i_p$, then the sum of the squares of all ...
17
votes
3answers
793 views

Eigenvalues of $A$ and $A + A^T$

This question has popped up at me several times in my research in differential equations and other areas: Let $A$ be a real $N \times N$ matrix. How are the eigenvalues of $A$ and $A + A^T$ related? ...
17
votes
4answers
1k views

The arithmetic-geometric mean for symmetric positive definite matrices

A while back, I wanted to see if the notion of the arithmetic-geometric mean could be extended to a pair of symmetric positive definite matrices. (I considered positive definite matrices only since ...
17
votes
2answers
2k 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" ...
17
votes
1answer
974 views

Proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ in Penrose graphical notation

For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation. Here $\det$ denotes the ...
16
votes
7answers
748 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 ...
16
votes
9answers
1k 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 ...
16
votes
5answers
957 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 ...
16
votes
4answers
1k 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 ...
16
votes
4answers
8k 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
3answers
8k views

Eigenvalues for the rank one matrix $uv^T$

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of ...
16
votes
2answers
866 views

$AB=BA$ implies $AB^T=B^TA$.

I am looking for an elementary proof (if such exists) of the following: $$ AB=BA \quad\Longrightarrow\quad AB^T=B^TA, $$ where $A$ and $B$ are $n\times n$ real matrices, and $A$ is a normal matrix, ...
16
votes
4answers
1k views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
16
votes
3answers
1k 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 ...