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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13
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1answer
284 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
13
votes
3answers
537 views

when does $\det(AB^T+BA^T)\le \det(AA^T+BB^T)$ hold?

When does the following matrix inequality hold? $$\det(AB+B^TA^T)\le \det(AA^T+BB^T)$$ $A$ and $B$ are any real matrices. My reply gives a counter example. The question is under what condition ...
13
votes
1answer
828 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 $v$ ...
13
votes
1answer
5k views

Generalized rotation matrix in N dimensional space around N-2 unit vector

There is a 2d rotation matrix around point $(0, 0)$ with angle $\theta$. $$ \left[ \begin{array}{ccc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right] $$ Next, ...
13
votes
1answer
9k 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 ...
13
votes
1answer
642 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
13
votes
3answers
2k 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 ...
13
votes
1answer
901 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 $c_{ij} \neq 0\ \forall i,j$): $$\left(\...
13
votes
2answers
622 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 & \...
13
votes
0answers
433 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 ...
13
votes
1answer
440 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 ...
13
votes
1answer
284 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
5answers
13k views

If $A^2 = I$ (Identity Matrix) then $A = \pm I$

So I'm studying linear algebra and one of the self-study exercises has a set of true or false questions. One of the question is this: If $A^2 = I$ (Identity Matrix) Then $A = \pm I$ ? I'm pretty ...
12
votes
6answers
1k views

A matrix satisfying $AB-BA=B$

If $A$ and $B$ are two matrices of $\mathcal{M}_n(\mathbb{R}$) such that $$AB-BA=B$$ how can we prove that $B$ isn't invertible? my attempt: I found that $\mathrm{tr}(B)=0$ but I know that this is ...
12
votes
4answers
14k views

Why are nonsquare matrices not invertible?

I have a theoretical question. Why are non-square matrices not invertible? I am running into a lot of doubts like this in my introductory study of linear algebra.
12
votes
4answers
1k views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ $$...
12
votes
5answers
3k views

To prove Cayley-Hamilton theorem, why can't we substitute $A$ for $\lambda$ in $p(\lambda) = \det(\lambda I - A)$?

Cayley Hamilton Theorem states that if $A$ is an $n \times n$ matrix over the field $F$ then $p(A) = 0$. We note that $p(\lambda) = \det(\lambda I - A)$. Hence, why can't we just substitute $\lambda$ ...
12
votes
7answers
2k 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 ...
12
votes
4answers
1k views

Does the inverse of the matrix always rely on the determinant of a matrix?

I always thought that if the determinant of a matrix $A$ is $0$ then it has no inverse, $(A^{-1})$, until I saw an exercise in Contemporary Abstract Algebra by Gallian. This asks me to prove that the ...
12
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 ...
12
votes
2answers
364 views

Prove that a symmetric matrix with a positive diagonal entry has at least one positive eigenvalue

Let $A$ be a symmetric martix $n \times n$ such that there is some $i$ such that $a_{ii}>0$. Prove that $A$ has a positive eigenvalue. I have a hint which I don't how to use/check: "Check ...
12
votes
6answers
390 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 ...
12
votes
3answers
362 views

Is there a name for matrix product with reversed indices?

The typical matrix product is as follows: $$ (\mathbf{A}\mathbf{B})_{ij} = \sum_{k=1}^m A_{ik}B_{kj}\,. $$ Is there a name or characterization for one such as $$(\mathbf{A}\mathbf{B})_{ij} = \sum_{...
12
votes
4answers
359 views

Is it true that all matrices in $M_2(\mathbb R)$ is the sum of two squares?

I recently show that every polynomial with real coefficient and $P$ is always positive. is a sum of two squares of polynomials. These questions also appear often in arithmetic. What if we change ...
12
votes
4answers
12k 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
3answers
65k 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
4answers
522 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
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. ...
12
votes
4answers
19k views

Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$

Given a matrix $A$ and column vector $x$, what is the derivative of $Ax$ with respect to $x^T$ i.e. $\frac{d(Ax)}{d(x^T)}$, where $x^T$ is the transpose of $x$? Side note - my goal is to get the ...
12
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 ...
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 $x_{j,k}=(j-1)\mathbin{\...
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\\ \end{...
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
373 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
810 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
253 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 v=(\det(A)^2-2\det(A)\...
12
votes
2answers
3k views

Coefficients of characteristic polynomial of a matrix

For a given matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$. If the characteristic polynomial of $A$ is $x^n+...
12
votes
2answers
7k 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
639 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
128 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
135 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 SL(...
12
votes
2answers
543 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 $C$, $C^k=(c_{ij})_{...
12
votes
1answer
320 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
874 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
163 views

Fake proofs using matrices

Having gone through the 16-page-list of questions using the tag (fake-proofs), and going though Best Fake Proofs? (A M.SE April Fools Day collection) and https://en.wikipedia.org/wiki/...
12
votes
1answer
262 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
132 views

Let $A$ and $x$ be $n \times n$ and $n \times 1$ matrices, all entries real and strictly positive. Assume that $A^2 x = x$. Show that $A x = x$.

I am stuck for weeks with the following problem: Let $A$ and $x$ be $n \times n$ and $n \times 1$ matrices, respectively, with all entries real and strictly positive. Assume that $A^2 x = x$. Show ...
12
votes
2answers
94 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 $V\in R^{d\times d}$. ...