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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12
votes
1answer
629 views

Detecting symmetric matrices of the form (low-rank + diagonal matrix)

Let $\Sigma$ be a symmetric positive definite matrix of dimensions $n \times n$. Is there a numerically robust way of checking whether it can be decomposed as $\Sigma = \mathcal{D} + v^t.v$ where $v$ ...
12
votes
2answers
520 views

The infinite-dimensional limit of sequence of solutions of linear equations when the number of equations goes to infinity

Suppose we have an infinite-dimensional real vector $y=(y_1,...)$. Suppose we have an infinite-dimensional real matrix $C=(c_{ij})$, $i,j\in\mathbb{N}$. Let $C^k$ be a submatrix of $C$, ...
12
votes
2answers
707 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
12
votes
2answers
441 views

Does this expression have a (“better”) determinant form?

[Edit I've found a determinant that satisfies the letter of the previous version of this question, but not its "Cayley-Menger" spirit.] A tetrahedron with face areas $w$, $x$, $y$, $z$ and ...
12
votes
1answer
417 views

Convexity of Matrix Exponential

Consider the function $A: \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$ defined as $$ A( x ) := \left[ \begin{matrix} x_1 & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & x_2 & ...
12
votes
1answer
336 views

A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
12
votes
1answer
216 views

Characterization of Volumes of Lattice Cubes

Here is a problem that came up in a conversation with a professor. I do not know if he knew the answer (and told me none of it) and has since passed so I can no longer ask him about it. Let $C$ be a ...
12
votes
1answer
266 views

Upper bound for the widest matrix with no two subsets of columns with the same vector sum

Over at PPCG there is an ongoing contest going on to find the largest matrix without a certain property, called property $X$. The description is as follows (copied from the question). A circulant ...
12
votes
3answers
752 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
11
votes
6answers
1k views

Matrix to power $2012$

How to calculate $A^{2012}$? $A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$ How can one calculate this? It must be tricky or something, cause there ...
11
votes
5answers
335 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.
11
votes
4answers
6k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
11
votes
3answers
3k views

Rank of skew-symmetric matrix

Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all ...
11
votes
6answers
367 views

Given a matrix $A$, find $A^n$

Given the matrix $$A = \left[{9\atop20}{-4\atop-9}\right]$$ how do I find $A^7$ or $A^{54}$ or $A^{2008}$ (etc.) ? I know I need the eigenvalues of A, but I'm not sure what to do afterwards. Is the ...
11
votes
4answers
3k views

What is the fastest way to find the characteristic polynomial of a matrix?

Finding the characteristic polynomial of a matrix of order $n$ is a tedious and boring task for $n > 2$. I know that: the coefficient of $\lambda^n$ is $(-1)^n$, the coefficient of ...
11
votes
1answer
2k views

Is the matrix $A$ diagonalizable if $A^2=I$

If $A$ is an involutory matrix, i.e. $A^2=I$, then is $A$ diagonalizable?
11
votes
3answers
2k views

$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.

The problem is from Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful. Let $A,B$ be $m \times n$ and $n ...
11
votes
3answers
2k views

What's the meaning of the transpose? [duplicate]

I don't understand the motivation of the transpose (or better yet, I haven't even seen one). It feels like just something pulled out of a hat. Thinking about it makes it seem like a product of being ...
11
votes
4answers
609 views

What makes elementary row operations “special”?

This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" ...
11
votes
2answers
4k views

Looking for insightful explanation as to why right inverse equals left inverse for square invertible matrices

The simple proof goes: Let B be the left inverse of A, C the right inverse. C = (BA)C = B(AC) = B This proof relies on associativity yet does not give any insight as to why this surprising fact ...
11
votes
6answers
295 views

Maximizing the sum $\sum\limits_{i=1}^nx_ix_{i+1}$ subject to $\sum\limits_{i=1}^nx_i=0$ and $\sum\limits_{i=1}^nx_i^2=1$

Is there an efficient way of solving the following problem? Given $x_i\in \mathbb R$, and that $\sum\limits_{i=1}^nx_i=0$ and $\sum\limits_{i=1}^nx_i^2=1$. I want to maximize ...
11
votes
4answers
516 views

Finding all matrices $B$ such that $AB=BA$ for a fixed matrix $A$

Let $$ A=\begin{pmatrix} 1 & 0 & 0 \\ 0& 1 & 0 \\ 3 & 1 & 2 \end{pmatrix} $$ Find all matrices $B$ such that $AB=BA$. Attempt at solution: I can show that $A$ is ...
11
votes
2answers
491 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
429 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
400 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
4answers
6k views

Is there a 3-dimensional “matrix” by “matrix” product?

Is it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form? I would appreciate it if you could help me to find out some tutorials online or mathematical ...
11
votes
2answers
1k views

Block Diagonal Matrix Diagonalizable

I am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diaginalizable, if only if $A$ and $B$ are diagonalizable. If $A\in\mathbb{C}^n$ ...
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
127 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
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
170 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
1answer
149 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
1answer
241 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
476 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
268 views

Prove that if $f$ is continuous at $0$, it is continuous on $\mathbb{R}$

Long story short, the question I'm stuck on is as follows: Let $f$ be a positive-definite function. Prove that if $f$ is continuous at $0$, then it is continuous everywhere. Here's the long ...
11
votes
2answers
444 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
3answers
217 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
210 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
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
284 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
174 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
617 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
714 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
345 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
481 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
674 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
5answers
839 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
3answers
1k views

Trace of powers of a nilpotent matrix

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be 0, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like to ...
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.