# Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

36,861 questions
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### Orbit of a symmetric matrix under orthogonal conjugation

Let $A\in M_n(\Bbb{R} )$ be a symmetric matrix. I want to find a general formula for the diagonals of the matrices of the form $g^{-1}Ag$, where $g\in O_n(\Bbb{R})$. Here is what I did : Since $A$ is ...
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### Is this simple symmetric matrix positive semi-definite?

Let the $n\times n$ symmetric matrix $A$, where $n\geq 9$ be given by A_{i,j}= \begin{cases} 1.4, &\text{for } 1\leq i=j\leq 9\\ (0.9)^{|i-j|},&\text{for } 1\leq i\neq j\leq 9\...
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### Eigenvalues and eigenvectors for the moment of inertia matrix

Find the eigenvalues and eigenvectors for the moment of inertia matrix given by $$I={m\over 2}\left(\begin{matrix} 1 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2\end{matrix}\right)$$ ...
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### On determinants and common divisors

Let $n\in\mathbb N$ and let $a_1,\ldots,a_n$ be natural numbers smaller than $10^n$. Write each $a_k$ in base $10$ and add $0$'s to the left of each decimal expansion, if needed, so that each $a_k$ is ...
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### Show $N$ is a normal subgroup of $G$ where $G$ is a invertible $2 \times 2$ matrix.

We have $G= \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \text{with$a$and$c$in$\{\pm 1\}$and$b$in$\mathbb{Z}$} \right\}$, which is given to be a subgroup of the group of ...
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### Matrix Equation AB = C with constraints, where both A and B are unknown.

As the title reads, I have the matrix equation: $$AB = C$$ With the constraint that all elements in B are greater than $0$ and less than $1$, and the last element of each row is $1$. A and B can be ...
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### Find the values of a and b that make the equation system have infinite solutions

I'm given the following system of equations and I need to find the values for a and b so that the system has infinite solutions. \begin{cases} &3y &- &2z &= 5 \\ &-2x &+ &...
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### Given path of a unit vector can we derive the rotation matrix?

Let's say a unit vector $\overrightarrow{A}(t)$ in N dimensions is continuously rotated around the origin. It may for example trace out a circle in 3 dimensions or some sort of spiral in higher ...
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### Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?
Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
Let $A \in \mathbb{R}^{n \times p}$, let $D$ be a diagonal matrix with positive entries. $\dagger$ denotes the Moore-Penrose pseudoinverse. Is it true in general that: (A^\top D A)^\dagger A^\top D ...