Articles of matrices

Solve $XA + A^T = I$ for $X$

I want to solve $$X \cdot A + A^T = I$$ for $X$, $A$ and $X$ are arbitrary matrices and $A$ is invertible. I know that $A \cdot A^{-1} = I$, this helps, but I don’t know how to deal with the additional $+A^T$. How can I approach this?

Computing the Frobenius normal form

I was wondering whether someone could give me an example how one actually determines the Frobenius normal form of a given matrix. Further, it seems hard to find an example where the new basis is calculated so that a given matrix is in Frobenius normal form. I really tried to find an example where this […]

Question on Smith normal form and isomorphism

Put $A=\begin{pmatrix} 1 & -5 & 4\\ 1 & -2 & 13\\ -2 & 13 & 7 \end{pmatrix}.$ The smith normal form of this matrix is \begin{pmatrix} 1 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 6 \end{pmatrix} and now I want to find $a , b, c$ $\in$ $\Bbb […]

Proving that all eigenvalues are $0$ bar one.

A symmetric $n\times n$ matrix is given by $\mathbf{A}=\lambda \mathbf{ee}^{\text{T}}$ where $\mathbf{e}$ is a unit vector. Show that $\mathbf{A}$ has an eigenvector of $\mathbf{e}$ with eigenvalue $\lambda$ and that all $n-1$ other eigenvalues are $0$. Working: $\mathbf{Ae}=\lambda \mathbf{e}(\mathbf{e}^{\text{T}}\mathbf{e})=\lambda\mathbf{e}$ so the first part is satisfied. I am stuck on showing that all other eigenvalues are $0$. […]

$A^2=cA$ for some $c \neq 0$

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $$A^2 = cA.$$ Questions. 1) Is there a matrix class for matrices those have this special property? If there is what is the name of it? 2) It is true or not, that […]

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 […]

A question on commutation of matrices

Given a diagonal matrix $D$, and a nilpotent matrix $N$, do we always have $DN=ND$? If not so, what further conditions do we need to have it? This question came form an ODE/Linear Algebra problem: Give $A\in \mathcal{M}_{n\times n}(\mathbb{R})$, there exists an invertible matrix $P$, and a matrix $B=D+N$, $D$ diagonal and $N$ nilpotent, such […]

How to convert the trace of this matrix expression to a quadratic form in terms of $b^2$?

i want to convert the following to a quadratic form in terms of vector $b$: $$trace(A^\top (\sum{b_i^2K_i})^\top T(\sum{b_i^2K_i}A))$$ where $T,K_i$ are symmetric and $n\times n$. A is $n\times m$ and $b_i$ is scalar. Can i write it in a quadratic form in terms of the vector $b$? or even $b^2$? if so, Then what would […]

Inverse of matrix of ones + nI

Having a vector $\mathbf{1} \in \mathbb{R}^{n}$ containing only ones, following equality should be true according to a paper I am currently reading: \begin{equation} \left( nI+\mathbf{1}\mathbf{1}^T \right)^{-1}= \frac{1}{n}\left( I – \frac{1}{2n} \mathbf{1}\mathbf{1}^T \right) \end{equation} EDIT: what is the general rule for constructing an inverse of a matrix with $n$ on diagonal and $1$ elsewhere and how […]

Solving Linear Systems with LU Decomposition and complete pivoting; stupid question

Given a matrix A and vector B, solve $Ax=B$ Using LU Decomposition with full Pivoting; $PAQ=LU$ where P and Q are row and column permutation vectors (correct me if I’m wrong) What I don’t understand is what to do with the permutation matrices to finish the solution. I know in partial pivoting, its simple $Lz=PB$ […]