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
4answers
11k views

Derivative of determinant of a matrix

Good morning everyone, I would like to know how to calculate: $\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big)$ Help me please. Thank you
12
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} = ...
12
votes
3answers
60k 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.
12
votes
7answers
7k views

Relation between Cholesky and SVD

When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed). Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
12
votes
4answers
506 views

How to prove that $A^2$ is a symmetric matrix

Conjecture 1 : Let $A$ be a real matrix such that $A^5=A A^T A A^T A$. Then $A^2$ is a symmetric matrix. (here $A^T$ denotes the transpose of a matrix A). I guess that the following is also ...
12
votes
2answers
1k views

Why are the eigenvalues of these “bitwise XOR matrices” integers?

In the course of playing around with this question, I have hit upon a question of my own. Consider the $n\times n$ symmetric matrix $\mathbf X$ whose entries are given by ...
12
votes
4answers
1k views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ ...
12
votes
2answers
1k views

Why is the identity the only symmetric $0$-$1$ matrix with all eigenvalues positive?

While thinking about this question I managed to convince myself that the identity is the only symmetric $0$-$1$ matrix with all eigenvalues positive. However, the argument is rather low-level. It ...
12
votes
3answers
341 views

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
12
votes
2answers
724 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 ...
12
votes
2answers
360 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
12
votes
2answers
767 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
12
votes
1answer
252 views

Find eigenvalues of unspecified matrix

Find all possible eigenvalues of a $2\times 2$ matrix $A$ satisfying $$\det(A^2)I-2\det(A)A+A^2=0.$$ Well, if $Av=\lambda v$ then $$\det(A^2)v-2\det(A)\lambda v+\lambda^2 ...
12
votes
1answer
2k views

What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
12
votes
2answers
6k 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 ...
12
votes
2answers
637 views

Properties of 4 by 4 Matrices

Define $ A=\begin{pmatrix} x_1 & x_2 & 0 & 0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 0 & 0 & 0\\ x_3 ...
12
votes
2answers
274 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 ...
12
votes
1answer
126 views

Is every multiplicative function from the matrix ring $\text{Mat}_n(R)$ to $R$ a function of $\text{det}$?

Suppose that $R$ is a commutative ring with $1$, and for $n \in \mathbb{N}$, let $\text{Mat}_n(R)$ be the set of $n \times n$ matrices with entries in $R$. It is well known that the determinant ...
12
votes
1answer
132 views

Check membership in a matrix group

I'm looking for a (preferably somewhat efficient) algorithm for this problem: Given a normal subgroup of $SL(m, \mathbb{Z})$ generated by a finite set $\{M_1, M_2, \dotsc, M_n\}$, and some $A \in ...
12
votes
1answer
798 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
538 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
1answer
310 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 ...
12
votes
2answers
857 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
1answer
256 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
0answers
151 views

Integrating a matrix function involving a determinant and exponential trace

I am trying to find the normalizing constant for a probability distribution and ran into a difficult integral. When $X$ is an $p \times k$ matrix, $a>0,$ and $g>0,$ how can I compute $$\int ...
12
votes
2answers
82 views

How to get the SVD of $2AA^T-\operatorname{diag}(AA^T)$ given $A$ and its SVD $A=USV^T$?

Given a matrix $A\in R^{n\times d}$ with $n>d$, and we can have some fast ways to (approximately) calculate the SVD (Singular Value Decomposition) of $A$, saying $A=USV^T$ and $V\in R^{d\times d}$. ...
12
votes
0answers
395 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
12
votes
1answer
347 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
0answers
327 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
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
14k views

Prove that if $AB$ is invertible then $B$ is invertible.

I know this proof is short but a bit tricky. So I suppose that $AB$ is invertible then $(AB)^{-1}$ exists. We also know $(AB)^{-1}=B^{-1}A^{-1}$. If we let $C=(B^{-1}A^{-1}A)$ then by the invertible ...
11
votes
5answers
367 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
3answers
5k 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
4answers
5k views

How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?

Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
11
votes
4answers
3k 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
7answers
14k 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 ...
11
votes
4answers
6k 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 ...
11
votes
3answers
566 views

Why is the matrix multiplication defined as it is? [duplicate]

Matrix multiplication is defined as: Let $A$ be a $n \times m$ matrix and $B$ a $m\times p$ matrix, the product $AB$ is defined as a matrix of size $n\times p$ such that $(AB)_i,_j = ...
11
votes
3answers
3k 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
3answers
1k views

Is this possible? AB- BA=I [duplicate]

I have just started linear functionals when I faced the following problem: If $A$ and $B$ are $n \times n$ complex matrices, show $AB - BA=\Bbb{I}$ is impossible. Can someone help me?
11
votes
4answers
663 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
5k 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
3answers
2k 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 ...
11
votes
6answers
309 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
1answer
5k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
11
votes
4answers
1k 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
1k 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 ...
11
votes
3answers
81 views

Shared eigenvectors between $A$ and $A^k$

$\newcommand\la{\lambda}$ Thanks to the spectral mapping theorem, we know that if $\la_1,\ldots,\la_n$ are the eigenvalues of a $n\times n$ complex matrix $A$, then $\la_1^k,\ldots,\la_n^k$ are the ...
11
votes
5answers
458 views

Why are there $736$ matrices $M\in \mathcal M_2(\mathbb{Z}_{26})$ 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
3answers
1k views

Matrix/Vector Derivative

I am trying to compute the derivative:$$\frac{d}{d\boldsymbol{\mu}}\left( (\mathbf{x} - \boldsymbol{\mu})^\top\boldsymbol{\Sigma} (\mathbf{x} - \boldsymbol{\mu})\right)$$where the size of all vectors ...