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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56
votes
19answers
5k views

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
54
votes
8answers
4k views

Why does this “miracle method” for matrix inversion work?

Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below: Problem: Let $A$ be a matrix ...
53
votes
9answers
80k 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 ...
53
votes
0answers
3k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
48
votes
5answers
28k 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?
48
votes
3answers
1k 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 ...
47
votes
12answers
10k 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 ...
47
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 = ...
41
votes
14answers
3k 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 ...
40
votes
4answers
4k views

Must eigenvalues be numbers?

This is more a conceptual question than any other kind. As far as I know, one can define matrices over arbitrary fields, and so do linear algebra in different settings than in the typical ...
40
votes
8answers
5k 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 ...
38
votes
4answers
3k 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 ...
38
votes
5answers
22k 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 ...
37
votes
6answers
46k views

Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric? I seem to remember a proof similar to this from my linear algebra class, ...
35
votes
8answers
54k 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 = ...
35
votes
2answers
2k 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 ...
35
votes
1answer
759 views

A System of Matrix Equations (2 Riccati, 1 Lyapunov)

Setup: Consider the following system of real matrices, \begin{align} {\bf P} &={\bf F}({\bf I}_{n}-{\bf K} {\bf H}^\top){\bf P}{\bf F}^\top+{\bf Q}, \;\;\;\;\;\;\;\;\text{where}\;\;\;\;{\bf ...
34
votes
5answers
28k 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\\ ...
33
votes
6answers
34k 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 ...
32
votes
3answers
18k views

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
32
votes
3answers
5k 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 ...
31
votes
3answers
2k views

How to find the determinant of this matrix?

Today at my linear algebra exam, there was this question that I couldn't solve. There was a matrix $A$ $$A=\begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & ...
31
votes
11answers
48k views

What is the usefulness of matrices?

I have matrices for my syllabus but I don't know where they find their use. I even asked my teacher but she also has no answer. Can anyone please tell me where they are used? And please also give me ...
30
votes
1answer
835 views

How prove this matrix $\det (A)=\left(\frac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}\neq 0$

Question: let $a_{i}>1,i=1,2,3,\cdots,n$,and such $a_{i}\neq a_{j}$,for any $i\neq j$ define the matrix $$A=\left(\dfrac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}$$ show that: ...
30
votes
2answers
2k views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : ...
29
votes
10answers
3k 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 ...
29
votes
5answers
20k views

Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero ...
29
votes
7answers
3k 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 ...
28
votes
1answer
900 views

Prove the determinant of this matrix

We have an $n\times n$ square matrix $\left(a_{i,j}\right)_{1\leq i\leq n, \ 1\leq j\leq n}$ such that all elements on main diagonal are zero, whereas the other elements are defined as follows: ...
27
votes
8answers
2k 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?
27
votes
2answers
11k views

Matrix multiplication: interpreting and understanding the process

I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ It ...
27
votes
3answers
669 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 ...
27
votes
5answers
1k views

What happens if I repeatedly alternately normalize the rows and columns of a matrix?

Here is an algorithm: input matrix M (in-place) divide each row of M by its norm divide each column of M by its norm repeat What will M look like after this has been repeated many times? Can we ...
0
votes
1answer
24 views

If $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many solutions

I was reading this pdf: https://www.math.ohiou.edu/courses/math3600/lecture10.pdf and it tells you that if $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many ...
0
votes
0answers
22 views

Operator norm of a diagonal matrix

I want to prove that the operator norm of a diagonal matrix $D$ is less than or equal to its largest value. I've tried the following but I don't know if it is correct. ...
0
votes
0answers
13 views

Upper bound on the norm of the inverse of matrices with zero limit

Let $\{L(\sigma)\}_{\sigma}$ be a family of matrices indexed by the parameter $\sigma$ so that the operator norm $||.||$ of $\{L(\sigma)\}_{\sigma}$ satisfies $Ae^{-a/\sigma}\leq ||L(\sigma)|| \leq ...
0
votes
0answers
16 views

Identifying Symmetric Matricies

If $A$ and $B$ are arbitrary $nxn$ matrices, which of the matrices must be symmetric? $something^T$ means the transpose of "something". $A^T$$A$ $A-A^T$ $A^TB^TBA$
0
votes
0answers
21 views

Identity involving pseudoinverses (Moore-Penrose) of symmetric matrices

Let A be a symmetric $m$ x $m$ matrix of rank r, and B a symmetric $m$ x $m$ matrix of rank $m - r$, such that $AB = 0$. Show that $A^+A+B^+B=I$, where $A^+$ is the Moore-Penrose pseudoinverse of $A$. ...
0
votes
1answer
21 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
0
votes
1answer
18 views

Lower bound for the distance between matrices of different rank.

This is a follow up question to this: Norm of diference of matrices of different rank Suppose $A$ is a norm one $n\times n$ matrix of rank $k$. What is $\inf\|A-B\|$, where the infimum is taken over ...
0
votes
0answers
16 views

A Projection Matrix which projects to a space $V$ with $dimV=2$ has $3$ eigenvalues which span a space of dimension=3

I have found an exercise involving a $3x3$ projection matrix which projects to a space $V$ with $dim(V)=2$. The matrix(or operator) is defined as $P=v*(v, v)+u*(u, u)$. So, in my understanding it ...
0
votes
0answers
39 views

Is there a variant of Euler's theorem for matrices?

Euler's theorem says that whenever $a$ and $p$ are coprime integers, then $a^{\phi(p)} \equiv 1 \mod p$ This is particularly useful for evaluating large powers of $a$: $a^n \ \%\ p = a^{n\ \%\ ...
0
votes
0answers
29 views

Divisibility of dimension by matrix equations.

Let ${\bf M}$ and ${\bf N}$ be two real $k\times k$ matrices such that ${\bf M}^2+{\bf N}^2={\bf M}{\bf N}$. Show that if $\det\left({\bf N}{\bf M}-{\bf M}{\bf N}\right)\neq 0$, then $3\mid k$.
0
votes
1answer
31 views

Prove: If $[T]$ is nilpotent of degree $k$, a $v$ exists such that $\{v,T(v),T^2(v),…,T^{k-1}(v)\}$ is linearly independent

I really need a hint on this one. I tried to look at nilpotent matrices but could not find a v that satisfies.
0
votes
2answers
42 views

How to prove that the square matrix $A_{n}$ matrix is nilpotent such that $A^{(n-1)}=0$

The matrix A looks like this: $$A=\begin{bmatrix} 0 & 1 & 0 & 0 & .&.&. &0\\ 0 & 0 & 2 & 0 & .&.&. &0\\ 0 & 0 & 0 & 3 ...
0
votes
1answer
999 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
0
votes
1answer
901 views

LU decomposition with row pivoting

Okay so consider a matrix $$ A = \left( \begin{array}{ccc} 0 & 1 & 1 \\ 1 & -2 & -1 \\ 1 & -1 & 1 \end{array} \right)$$ so obviously to get the upper and lower traiangular ...
-1
votes
0answers
11 views

Solving an initial value problem consisting of a matrix. Please need help!!

I am doing differential equations problems and the problem states this.... Solve the initial value problem \begin{align} X'(t) &= \begin{pmatrix} 1 & -2 & 2 \\ -2 & 1 & -2 \\ 2 ...
-1
votes
0answers
20 views

Setting up Vandermonde matrix with data points containing derivatives

I know how to setup the Vandermonde matrix with simple interpolation conditions, such as p(-1)=4 p(0)=7 p(1)=28 p(2)=247, i.e. (-1,4),(0,7),(1,28),(2,247) ...
-9
votes
1answer
236 views

Showing that a matrix is symmetric positive definite [on hold]

I am trying to show that, if the SPD matrix $$K = \begin{bmatrix} A & B \\ B^T & C \\ \end{bmatrix} \in \mathbb{R}^{n \times n},$$ where $A \in \mathbb{R}^{m \times m}$ is also SPD, then ...