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
2answers
566 views

Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial ...
11
votes
3answers
436 views

Eigenvalues of some peculiar matrices

While I was toying around with matrices, I chanced upon a family of tridiagonal matrices $M_n$ that take the following form: the superdiagonal entries are all $1$'s, the diagonal entries take the form ...
11
votes
5answers
414 views

Why are there$ 736$ $2\times 2$ matrices $(M)$ over $\mathbb{Z}_{26}$ for which it holds that $M=M^{-1}$?

I'm currently trying to introduce myself to cryptography. I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for ...
11
votes
1answer
483 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 ...
11
votes
2answers
11k 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 ...
11
votes
1answer
1k views

4 by 4 Matrix Puzzle

I was solving the puzzle for the Company interview exam. I found this puzzle, I cannot come up with the solution. How to solve it and what is the correct answer? Determine the number of $4\times ...
11
votes
1answer
140 views

Largest determinant of a real $3\times 3$-matrix

What is the largest determinant of a real $3\times 3$-matrix with entries from the interval $[-1,1]$ ? A result of John Williamson says that the largest value is equal to $4$, if the entries are just ...
11
votes
2answers
189 views

Prove or disprove : $\det(A^k + B^k) \geq 0$

This question came from here. As the OP hasn't edited his question and I really want the answer, I'm adding my thoughts. Let $A, B$ be two real $n\times n$ matrices that commute and $\det(A + ...
11
votes
2answers
4k views

Determine the matrix relative to a given basis

Question: (a) Let $f: V \rightarrow W$ with $ V,W \simeq \mathbb{R}^{3}$ given by: $$f(x_1, x_2, x_3) = (x_1 - x_3, 2x_1 -5x_2 -x_3, x_2 + x_3).$$ Determine the matrix of $f$ relative to the basis ...
11
votes
2answers
313 views

Given $BA$, find $AB$.

If $A, B$ are $4\times 3, 3\times 4$ real matrices, respectively, $$BA=\begin{bmatrix} -9 & -20 & -35 \\ 2 & 5 & 7 \\ 2 & 4 &8 \end{bmatrix}$$ $$AB=\begin{bmatrix} -14 & ...
11
votes
1answer
162 views

Prove a $n \times n $ matrix has rank 3

I have been examining a problem dealing with finding the rank of a $n \times n $ matrix $M$ as follows: \begin{bmatrix} 0&1&4&9&16&\cdots &(n-1)^2\\ ...
11
votes
3answers
581 views

Generation of unimodular matrices with bounded elements

Does anybody know what is the algorithm for generating random unimodular matrices (integer matrices with determinant $\pm 1$) whose elements do not exceed a given bound? Such an algorithm is mentioned ...
11
votes
1answer
2k views

Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to ...
11
votes
1answer
2k 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 ...
11
votes
2answers
580 views

Determinant of a finite-dimensional matrix in terms of trace

I have noticed that for the case of 1x1, 2x2 and 3x3 matrices $A$, $B$, I can write the determinant of their commutator $C=[A,B]$ in terms of traces: 1x1 matrices $A$, $B$: $$\det(C)=\text{tr}(C)$$ ...
11
votes
1answer
264 views

Proving a certain determinant $\left|\det A\right|$ is complete square

Consider the following matrix $$ A_{ij}= \begin{cases} 1\quad\text{ if }\space (i+j)\space\text{ is prime,}\\ 0\quad\text{ otherwise.} \end{cases} $$ How can one prove that $\left|\det A\right|$ is a ...
11
votes
2answers
481 views

Geometric interpretation of normal and anti-hermitian matrices

How do I interpret following types of matrices as special types of transformations? I mean what are the transformative properties of following types of matrices, from $\mathbb{R}^n $ to $ ...
11
votes
4answers
319 views

Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its ...
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
298 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
222 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
681 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
5k views

When does a Square Matrix have an LU Decomposition?

When can we split a square matrix (rows = columns) into it’s LU decomposition? The LUP (LU Decomposition with pivoting) always exists; however, a true LU decomposition does not always exist. How do ...
11
votes
1answer
775 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
384 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 ...
10
votes
4answers
702 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
4k 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
9answers
1k views

Matrix with zeros on diagonal and ones in other places is invertible

Hi guys I am working with this and I am trying to prove to myself that n by n matrices of the type zero on the diagonal and 1 everywhere else are invertible. I ran some cases and looked at the ...
10
votes
5answers
868 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
2answers
2k 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
1k 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
829 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
3answers
651 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
487 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
3answers
4k views

why determinant is volume of parallelepiped in any dimensions

for $n = 2,$ I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculate the area by coordinates. But how can one easily realize that it is true ...
10
votes
7answers
17k 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
2answers
2k views

Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant

If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$? I remember seeing such a ...
10
votes
3answers
1k 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
227 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
2k views

Can a basis for a vector space be made up of matrices instead of vectors?

I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that ...
10
votes
2answers
186 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
401 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
915 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
8k 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
289 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
5k views

The determinant of block triangular matrix as product of determinants of diagonal blocks

I am given the following partitioned - upper-triangular matrix: $$ \begin{bmatrix} A_1 &* &* &* &* &* \\ 0& A_2 &* &* &* &* \\ .& 0& ...
10
votes
2answers
7k 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
198 views

Find this Determinant

I have to find this determinant, call it $D$ \begin{vmatrix} \frac12 & \frac1{3}& \frac1{4} & \dots & \frac1{n+1} \\ \frac1{3} & \frac14 & \frac15 & \dots & ...