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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11
votes
1answer
180 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 ...
11
votes
1answer
288 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) = ...
11
votes
1answer
214 views

$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a ...
11
votes
1answer
178 views

How many arrays with crossed cells, order of rows/columns irrelevant

I've been struggling with this simple problem for months though as I am a newbie to… well, maths, there's high chance someone more educated than myself may get it right! Let's consider an array or a ...
11
votes
1answer
645 views

spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d entries with variance $\sigma^2/n$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$ ...
11
votes
1answer
736 views

Determinant of a symmetric matrix

Given an $n\times n$ matrix $C= [c_{ij}]$ which is symmetric (i.e. $c_{ij}=c_{ji}\ \forall i,j$) calculate the determinant of the following matrix (assume $c_{ij} \neq 0\ \forall i,j$): ...
11
votes
0answers
356 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 ...
11
votes
0answers
493 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 ...
10
votes
4answers
681 views

If $A$ is singular, is $A^3+A^2+A$ singular?

Suppose that $A$ is singular, is $A^3 + A^2 + A$ singular as well?
10
votes
6answers
3k 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. ...
10
votes
5answers
851 views

Matrix raised to 14th power

Calculate $\left(\begin{matrix} 6&1&0\\0&6&1\\0&0&6\end{matrix}\right)^{14}$ Whould I do it one by one, and then find a pattern? I sense $6^{14}$ on the diagonal, and zeroes ...
10
votes
6answers
7k views

Sum of all elements in a matrix

The trace is the sum of the elements on the diagonal of a matrix. Is there a similar operation for the sum of all the elements in a matrix?
10
votes
2answers
1k views

Show that $ e^{A+B}=e^A e^B$

If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that $$ e^{A+B}=e^A e^B$$ Note that $A$ and $B$ do NOT have to be diagonalizable.
10
votes
7answers
979 views

Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?

I saw the following statement in my homework and we are asked to make use of the statement: If $S$ is a symmetric matrix then $$\det(I + S ) = 1 + \operatorname{trace}(S).$$ However, I am not ...
10
votes
4answers
810 views

$AB \neq 0$ but $BA=0$

Do there exists to matrices or objects such that $AB \neq 0$ but $BA=0$? Another way to ask this question is if there exists objects or matrices $A$ and $B$ such that... $[A,B]=AB$ where $[ \, , \, ]$ ...
10
votes
7answers
10k 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 ...
10
votes
3answers
632 views

Proving or disproving A+B is invertible

Given two matrices $A,B\in M_n (F)$, where $A$ is $k$ -nilpotent and $B$ is invertible, is it true that $A+B$ is also invertible? I was having trouble on how to prove this, and then I thought maybe ...
10
votes
5answers
470 views

Prove $BA - A^2B^2 = I_n$.

I have a problem with this. Actually, still don't have the right way to start :/ Problem : Let $A$ and $B$ be $n \times n$ complex matrices such that $AB - B^2A^2 = I_n$. Prove that if $A^3 + B^3 = ...
10
votes
7answers
15k views

How to check if a symmetric $4\times4$ matrix is positive semi-definite?

How does one check whether symmetric $4\times4$ matrix is positive semi-definite? What if this matrix has also rank deficiency: is it rank 3?
10
votes
4answers
4k views

Solving very large matrices in “pieces”

Say you have a very dense matrix that is 30000x30000 elements. The very dense matrix comes from the radiosity equation, which I discussed here. Say you have Ax = B. You have B, and A is 30000x30000 ...
10
votes
3answers
892 views

Quick way to find eigenvalues of anti-diagonal matrix

If $A \in M_n(\mathbb{R})$ is an anti-diagonal $n \times n$ matrix, is there a quick way to find its eigenvalues in a way similar to finding the eigenvalues of a diagonal matrix? The standard way for ...
10
votes
5answers
214 views

Invertibility of a Kronecker Product

Prove that $A\otimes B$ is invertible if and only if $B\otimes A$ is invertible. I don't have a clue where to start to be honest. I am not very familiar yet to the Kronecker Product so could you ...
10
votes
1answer
1k 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} = ...
10
votes
2answers
181 views

Determinant of $4\times4$ Matrix

I tried to solve for a $4 \times 4$ matrix, but I'm unsure if I did this properly, can anyone tell me if I did this correct? Or if there were any mistakes where at? Also, I know this is an inefficient ...
10
votes
3answers
393 views

What is the largest determinant you can get by filling in 0,1 or 2 into a 4-by-4 matrix?

For example $$\left| \begin{array}{ccc} 2 & 0 & 0 & 2 \\ 2 & 0 & 2 & 0 \\ 0 & 2 & 1 & 2 \\ 2 & 2 & 0 & 0 \end{array} \right|=40$$ Can it get bigger ...
10
votes
2answers
887 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 ...
10
votes
2answers
280 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
10
votes
2answers
7k views

Reflection across a line?

The linear transformation matrix for a reflection across the line $y = mx$ is: $$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix} $$ My professor gave us the formula ...
10
votes
2answers
6k views

How to calculate the gradient of log det matrix inverse?

How to calculate the gradient with respect to $X$ of: $$ \log \mathrm{det}\, X^{-1} $$ here $X$ is a positive definite matrix, and det is the determinant of a matrix. How to calculate this? Or ...
10
votes
3answers
35k views

shortcut for finding a inverse of matrix

I need tricks or shortcuts to find the inverse of $2 \times 2$ and $3 \times 3$ matrices. I have to take a time-based exam, in which I have to find the inverse of square matrices.
10
votes
1answer
451 views

Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular

Given a random square matrix of size $n\times n$ in the field $\mathbb F_2$, what is the probability that its determinant is $1$? (This is also the probability that the matrix is non-singular, since ...
10
votes
2answers
10k views

Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks

Let $A$ be a block upper triangular matrix: $A = \left( \begin{matrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{matrix} \right)$ where $A_{1,1} ∈ C^{p×p}$, $A_{2,2} ∈ C^{(n-p)×(n-p)}$ Show ...
10
votes
1answer
819 views

Example of similar matrices $A$ and $B$ such that products $AB$ and $BA$ are not similar

I'm looking for the simplest possible example of square matrices $A$ and $B$ such that $A$ is similar to $B$, $AB$ is not similar to $BA$. Such an example should exist, but I would like to find ...
10
votes
4answers
34k views

Diagonalizable Matrices: How to determine?

I am trying to figure out how to determine the diagonalizability of the following two matrices. For the first matrix $$\left[\begin{matrix} 0 & 1 & 0\\0 & 0 & 1\\2 & -5 & ...
10
votes
2answers
2k views

Calculating RGB plus Amber

I'm currently working on a wide gamut light source using red, green and blue LED emitters. From an internal xyY (or CIE XYZ) representation, I can reach any color or color temperature via a 3x3 ...
10
votes
1answer
127 views

How many matrices in $M_n(\mathbb{F}_q)$ are nilpotent?

I have strong computational evidence to think that the answer is $q^{n(n-1)}$, although a proof eludes me. Any ideas?
10
votes
3answers
160 views

Determinant of $a_{i,j}=(x_i+y_j)^k$

How can I find the determinant of the matrix $A\in\mathcal{M}_n(\mathbb{R})$ with coefficients $a_{i,j}=(x_i+y_j)^k,k<n$ ? All the $x_u,y_u$ are real numbers. Derivating won't help, and I didn't ...
10
votes
3answers
231 views

Show that $A$ is symmetric, with $A \in M_n(\mathbb R)$

Let $A \in M_n(\mathbb R)$. Show that if $A(\,{}^t\!A)A$ is is symmetric, then $A$ is also symmetric. My attempt: If $A \in Gl_n(\mathbb{R})$, We have : ${}^t(A^{-1})=(\,{}^t\!A)^{-1}$ ...
10
votes
3answers
229 views

Existence of some type matrix

Is there square matrix $A$ of size $3$ with real entries such that $$ \operatorname{tr}(A)=0\text{ and }A^2+A^T=I. $$ I have proved that there is not with size $2$ using definition of "trace", but ...
10
votes
2answers
450 views

Characteristic polynomial of a matrix with zeros on its diagonal

Let $p(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+\cdots+a_1x+a_0=(x-\lambda_1)\cdots(x-\lambda_n)$ be a polynomial with real coefficients such that every $\lambda_i$ is real. Is there always a symmetric ...
10
votes
1answer
307 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 ...
10
votes
1answer
1k views

In-place inversion of large matrices

In Solving very large matrices in "pieces" there is a way shown to solve matrix inversion in pieces. Is it possible to apply the method in-place? I am refering to the answer in the ...
10
votes
1answer
1k views

Intuition for complex eigenvalues

The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same. My question What is the ...
10
votes
2answers
448 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
2answers
354 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
1answer
126 views

Could one make a ring of matrices of uncountable size?

I've seen several kinds of matrices. You could see a real square matrix as a mapping: $$ A \quad : \quad \{1, 2,\cdots, n \}^2 \ \longrightarrow \ \mathbb{R} $$ I've seen that there were also infinite ...
10
votes
1answer
3k views

Inverse of a block matrix

I have a special case where $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: $X$ is non-singular $A$ is singular $B$ is full column rank $C$ is full row rank How do you ...
10
votes
2answers
1k 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: ...
10
votes
2answers
147 views

$AB=z \mathrm{Id}_n$ implies $z^m BA = z^{m+1} \mathrm{Id}_n$ for what $m$?

This question builds on a series of questions looking for elementary proofs that $AB=\mathrm{Id}$ implies $BA=\mathrm{Id}$, for $A$ and $B$ both $n \times n$ matrices over a commutative ring. First ...
10
votes
1answer
240 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: ...