Articles of matrix equations

Tangent Space of SL(n,R) at arbitrary point, e.g. not at $\mathbb{1}$

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 […]

Solve the matrix equation $X = AX^T + B$ for $X$

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 $B$?

Is there anything like upper tridiagonal matrix? How to find the determinant of such a matrix?

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 […]

Eigenvectors of the $2\times2$ zero matrix

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 $\begin{bmatrix}1 \\ 0\end{bmatrix}$ […]

Understanding Gauss-Jordan elimination

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 […]

Set of all positive definite matrices with off diagonal elements negative

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. […]

An inequality on the root of matrix products

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 […]

Algebraic Riccati equation (DARE) stabilazability condition

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 […]

When does this matrix have zero, one and infinite solution?

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} […]

How to solve the matrix equation $ABA^{-1}=C$ with $\operatorname{Tr}(A)=a$

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 $N\times N$.