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), ...
49
votes
16answers
8k views
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$ where $I$ is identity matrix. Show that $BA = I$. I do not understand anything more than the following.
Elementary row operations.
Linear ...
41
votes
3answers
738 views
Alice and Bob matrix problem.
Alice and Bob play the following game with an $n*n$ matrix, where $n$ is odd.
Alice fills in one of the entries of the matrix with a real number. then Bob fills one. Then Alice and so on so forth ...
36
votes
14answers
2k views
'Linux' math program with interactive terminal?
Are there any open source math programs out there that have an interactive terminal and that work on linux?
So for example you could enter two matrices and specify an operation such as multiply and ...
36
votes
8answers
6k views
What does matrix multiplication actually mean?
If I multiply two numbers, say 3 and 5, I know it means add 3 to itself 5 times or add 5 to itself 3 times.
But If I multiply two matrices, what does it mean ?
I mean I can't think it in terms of ...
36
votes
2answers
1k views
Polynomial equations $p(A, B) = 0$ for matrices that ensure $AB = BA$
Let $k$ be a field with characteristic different from $2$, and $A$ and $B$ be $2 \times 2$ matrices with entries in $k$. Then we can prove, with a bit art, that $A^2 - 2AB + B^2 = O$ implies $AB = ...
26
votes
4answers
1k views
Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields
Some days ago, I was thinking on a problem, which states that $AB-BA=I$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ denotes ...
26
votes
6answers
1k views
How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?
I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its ...
24
votes
3answers
403 views
Square matrices satisfying certain relations must have dimension divisible by $3$
I saw this tucked away in a MathOverflow comment and am asking this question to preserve (and advertise?) it. It's a nice problem!
Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with ...
23
votes
7answers
1k views
Can the matrices $A$ and $I+A$ have the same determinant?
Let $A\in\mathbb R^{n\times n}$ be an arbitrary matrix. Can $A$ and $I+A$ have the same determinant, if not how to prove it?
Furthermore, can $A$ and $I+A$ have the same eigenvalues?
23
votes
7answers
1k views
How to tell if some power of my integer matrix is the identity?
Given an $n\times n$-matrix $A$ with integer entries, I would like to decide whether there is some $m\in\mathbb N$ such that $A^m$ is the identity matrix.
I can solve this by regarding $A$ as a ...
21
votes
3answers
406 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
2answers
220 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 ...
20
votes
2answers
344 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 ...
20
votes
2answers
481 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
4answers
356 views
Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected
Math people:
In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
17
votes
11answers
1k views
What is the usefulness of matrices?
I have matrices for my syllabus but I don't know where do they find their use. I even asked my teacher but she also has no answer. Can anyone please tell me where are they used?
And please also give ...
16
votes
4answers
580 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 $\det(A)=0$?
16
votes
3answers
571 views
Why, historically, do we multiply matrices as we do?
Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
16
votes
1answer
523 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" ...
16
votes
1answer
712 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 ...
15
votes
1answer
1k views
Is the following matrix invertible?
$$
\begin{bmatrix} 1235 &2344 &1234 &1990\\
2124 & 4123& 1990& 3026 \\
1230 &1234 &9095 &1230\\
1262 &2312& 2324 &3907
\end{bmatrix}$$
Clearly its ...
15
votes
4answers
1k views
Matrices which are both unitary and Hermitian
Matrices such as
$$
\begin{bmatrix}
\cos\theta & \sin\theta \\
\sin\theta & -\cos\theta
\end{bmatrix}
\text{ or }
\begin{bmatrix}
\cos\theta & i\sin\theta \\
...
15
votes
3answers
385 views
Matrices - Conditions for $AB+BA=0$
The Problem
Let $A$ be the matrix $\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix} \bigr)$, where no one of $a,b,c,d$ is $0$. Let $B$ be a $2\times 2$ matrix such that ...
15
votes
2answers
5k views
How do I tell if matrices are similar?
I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.
I solved this by using a matrix called $S$:
$$\left(\begin{array}{cc}
a& b\\
...
15
votes
4answers
896 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 ...
14
votes
4answers
1k 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 ...
14
votes
5answers
4k views
Importance of rank of a matrix
What is the importance of rank of a matrix ?
I know that rank of a matrix is the number of linearly independent rows/columns (whichever is smaller). Why is it a problem if a matrix is rank ...
14
votes
9answers
547 views
Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?
I am searching for a short coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ for linear operators $A$, $B$ between finite dimensional vector spaces of the same dimension. The ...
14
votes
5answers
601 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 ...
14
votes
5answers
459 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
$$A = \left[
\begin{matrix}
1 & a & a^2 & a^3 \\
a ...
14
votes
5answers
948 views
$\sin(A)$, where $A$ is a matrix
If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
14
votes
4answers
574 views
How does multiplying by trigonometric functions in a matrix transform the matrix?
I found this comic:
But I can't understand the humor because I can't understand how trig functions affect matrix multiplication. Can someone please explain?
14
votes
2answers
216 views
easier way of calculating the determinant for this matrix
I have to calculate the determinant of this matrix:
$$
\begin{pmatrix}
a&b&c&d\\b&c&d&a\\c&d&a&b\\d&a&b&c
\end{pmatrix}
$$
Is there an easier way of ...
14
votes
4answers
345 views
Powers of random matrices
Let $M$ be an $n \times n$ matrix whose elements are random reals in [0,1].
Two questions.
What is the growth rate of the magnitude of the elements of $M^k$ as a function of $k$? It is definitely
...
14
votes
4answers
436 views
Determinant of a generalized Pascal matrix
Let $M$ denote the infinite matrix defined recursively by
$$
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}, & ...
14
votes
1answer
126 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}
...
14
votes
1answer
252 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 ...
13
votes
9answers
1k 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 ...
13
votes
5answers
536 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 ?
13
votes
5answers
454 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 ...
13
votes
5answers
13k views
Inverse of the sum of matrices
I have two square matrices - $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case ...
13
votes
4answers
12k views
Inverse of a triangular matrix( both upper & lower ) is triangular
How can we prove that inverse of upper(lower) triangular matrix is upper(lower) triangular...
Can anybody answer to this question.......
Thanks in advance.........
13
votes
3answers
711 views
How to show determinant of a specific matrix is nonnegative
How to show that
$$\det A= \det \begin{pmatrix}\cos\frac{\pi}{n}&-\frac{\cos\theta_1}{2}&0&0&\cdots&0&-\frac{\cos\theta_n}{2}
...
13
votes
2answers
187 views
Why does calculating matrix inverses, roots, etc. using the spectrum of a matrix work?
Suppose $A$ is a $n \times n$ matrix from $M_n(\mathbb{C})$ with eigenvalues $\lambda_1, \ldots, \lambda_s$. Let $$m(\lambda) = (\lambda - \lambda_1)^{m_1} \ldots (\lambda - \lambda_s)^{m_s}$$
be the ...
12
votes
2answers
614 views
What's the name for the property of a function $f$ that means $f(f(x))=x$?
I can think of several examples of functions such that twice application of the function is equivalent to no application of it.
Additive inverse
Multiplicative inverse
Fourier transform
Complex ...
12
votes
5answers
293 views
Matrices with $A^3+B^3=C^3$
Problem: Find infinitely many triples of nonzero $3\times 3$ matrices $(A,B,C)$ over the nonnegative integers with
$$A^3+B^3=C^3.$$
My proposed solution is in the answers.
12
votes
4answers
384 views
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
I was reviewing some matrices and found this interesting
if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
12
votes
4answers
310 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.
12
votes
4answers
917 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 $p$ ...
12
votes
4answers
577 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 ...
