I am looking for the tangent space of $SL(n,\mathbb{R}) = \{A\in\mathbb{R}^{n\times n} | \det{A}=1 \}$ (actually $n=3$) at an arbitrary $M\in SL(n,\mathbb{R})$. From now on I will omit the $\mathbb{R}$ for convenience. It is well known and easy to prove that the tangent space at the identity matrix $\mathbb{1}$, $T_1SL(n)$, is the set of all […]

Consider the matrix equation \begin{equation} X=AX^T+B, \end{equation} where $X$ is an unknown square matrix. Is it possible to solve it analytically? Moreover, can a general solution be written down in terms of the matrices $A$ and $B$?

I want to find the determinant of the following matrix. $$\left[\begin{matrix} -\alpha_1 & \beta_2 & -\gamma_3 & 0 & 0 & 0 & \cdots & 0&0 \\ 0 & -\alpha_2 & \beta_3 & -\gamma_4 & 0 & 0 & \cdots & 0 & 0 \\0 & 0 & -\alpha_3 & \beta_4 & -\gamma_5&0&\cdots&0 & 0 […]

I have been given a problem that involves the following matrix: $$\begin{bmatrix}-2 & 0\\0 & -2\end{bmatrix}$$ I calculated the eigenvalues to be $\lambda_{1,2} = -2$ When I go to calculate the eigenvectors I get the following system: $$\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$$ The eigenvectors are clearly $\begin{bmatrix}1 \\ 0\end{bmatrix}$ […]

I have a following system: $$x_1 + x_2 – x_3 = 5$$ $$2x_1 + 2x_2 – 4x_3 = 6$$ $$x_1 + x_2 – 2x_3 = 3$$ I dont understand how to solve this system using Gauss-Jordan elimination. I was told it I had to have a final result of something like this: let $x$ represents […]

Please, help me to find the set of all positive definite matrices (PDM) of which off diagonal elements are negative. Considering the case with n=2, the symmetric mat $A=[a_1, a_2;a_2, a_3]$ needs to be a PD. i.e. $x’Ax>0$, $a_2\neq 0$, $x\neq 0$ where $x=[x_1,x_2]’$. Always we have $a_1>0$ and $a_3>0$ as they are principal minors. […]

Suppose $A$ and $B$ are positive definite (symmetric) real matrices. Is it true that $(AB)^{1/2}+(BA)^{1/2} \geq A^{1/2}B^{1/2}+B^{1/2}A^{1/2}$ ? EDIT: Shown to be false below. Extension sought: Consider now a square strictly contractive matrix $S$ (such that $I-SS^T>0$) and consider the inequality $(AB)^{1/2}+(BA)^{1/2}\geq A^{1/2}SB^{1/2}+B^{1/2}S^TA^{1/2}$ ? Does this contraction change things, i.e. is the inequality true? A […]

I’m Trying to help in this question which involves Algebraic Ricatti equation. Honestly to say I never met this equation before. I’m struggling to understand the conditions stated in the limitations of the dare algorithm in matlab. For convinience I quote it here: The $(A, B)$ pair must be stabilizable (that is, all eigenvalues of […]

So I’ve forgotten what the conditions for when a matrix has zero, one and infinitely many solutions. Starting with this matrix: $$ \begin{align} &\left[\begin{array}{rrr|r} 1 & 1 & -1 & 0 \\ 2 & -1 & -5 & 3\\ -1 & 2 & a^2+3a & -3a \end{array}\right] \end{align} $$ I reduced this to: $$ \left[\begin{array}{rrr|r} […]

I have the following matrix equation: $$ABA^{-1}=C$$ with $B$ and $C$ given and $A$ unknown. The constraint on $A$ is $\operatorname{Tr}(A)=a$ with $a\in\mathbb{R}$. The matrices are $N\times N$.

Intereting Posts

a theorem of Fermat
Suppose $p$ is a prime number and $a$ is an integer. Show that if $p \mid a^n$, then $p^n \mid a^n$ for any $n \geq 1$?
Convolute exponential with a gaussian
Determine $a$ values allowing $x^2+ax+2$ to be divided by $x-3$ in $\mathbb Z_5$
Prove every strongly connected tournament has a cycle of length k for k = 3, 4, … n where n is the number of vertices.
An inequality, which is supposed to be simple
Given that$(3x-1)^7=a_7x^7+a_6x^6+a_5x^5+…+a_1x+a_0$, find $a_7+a_6+a_5+a_4+…+a_1+a_0$
Construction of an infinite set such that any two number from the set are relatively prime
Formula for Sum of Logarithms $\ln(n)^m$
An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$
Orthonormal Hamel Basis is equivalent to finite dimension
Calculate $\underset{x\rightarrow7}{\lim}\frac{\sqrt{x+2}-\sqrt{x+20}}{\sqrt{x+9}-2}$
Splitting of primes in the compositum of fields
Dirac delta in polar coordinates
Linear Transformations on Function Spaces