Articles of matrices

Invertibility of $BA$

I’m having trouble with the following question which may seem simple but to me it’s not Let $A \in \mathbb{R}^{5 \times 7}, B\in \mathbb{R}^{7 \times 5}$. Prove that $BA$ is not invertible. I thought maybe use the relation to the solution sets of $Ax=0$ which is infinite, hence $A$ isn’t invertible. But I don’t know […]

If $A$ is normal and upper triangular then it is diagonal

Let $A$ be a normal matrix in Mat$_{n\times n}(\mathbb C)$, if $A$ is upper triangular then it is diagonal (Normal means $AA^*=A^*A$, where $A^*$ is the conjugate transpose of $A$) If I consider the diagonal of $AA^*$, let denote $(a_{ij})=A$ and $(â_{ij})_{i,j}=AA^*$ then, since $AA^*=A^*A$ $â_{ii}=\sum\limits_{k=1}^na_{ik}\overline{a}_{ik}=\sum\limits_{k=1}^n\overline{a_{ki}}{a}_{ki}$ $\implies\sum\limits_{k=1}^n|a_{ik}|^2=\sum\limits_{k=1}^n|a_{ki}|^2$. If I take $i=n$ then it follows that […]

Householder reflections

Let $x=\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ I want to use a Householder reflector U to keep only first element in vector x, and make everything else zero but I’m doing something wrong… $U=I-\frac{uu^T}{\beta}$ $\beta=\frac{\left \| u \right \|_2^2}{2}$ $Ux=x-u$ $\beta=\frac{16}{2}=8$ $u=\begin{bmatrix} 0\\ 2\\ 3 \end{bmatrix}$ $U=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & […]

Why doesn't the minimal polynomial of a matrix change if we extend the field?

Why doesn’t the minimal polynomial of a matrix change if we extend the field? I appreciate any help or proof.

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ Now the thing is that $Df$ is a $3 \times 2$ matrix, so I cannot invert this matrix easily. So how do […]

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the question from a) still true if $AB\neq BA$ ? AND SOLUTION: Since $AB=BA$, $A,B$ are simultaneously triangularizable. Thus $P(z)=det(A+zB)=(\lambda_1+z\mu_1)\times\cdots\times(\lambda_n+z\mu_n)$ where $spectrum(A)=(\lambda_i)_i,spectrum(B)=(\mu_i)_i$. $P$ is a polynomial s.t. $|P(z)|=1$ if $|z|=1$. […]

Is characterisation of degree 2 nilpotent matrices (i.e. $M^2=0$) known?

$M$ is $n\times n$ real (or complex) matrix. Also $M$ is nilpotent of degree 2, i.e. $M^2=0.$ Question. How does $M$ look like? I just calculated that $2\times 2$ matrix must have following form $$\begin{bmatrix} gh & \pm g^2 \\ \mp h^2 & -gh \end{bmatrix}.$$ I wanted to compute conditions on $3\times3,4\times 4$ and look […]

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix?

What is the difference between using $PAP^{-1}$ and $PAP^{T}$ to diagonalize a matrix? Can both methods be used to diagonalize a diagonalizable matrix $A$? Also does $A$ been symmetric or not effect which method to use?

What should be the characteristic polynomial for $A^{-1}$ and adj$A$ if the characteristic polynomial of $A$ be given?

Let the characteristic polynomial of $A$ be $\psi_A(x):=p(x)$. If $A$ be non-singular, then find that the characteristic polynomial of $A^{-1}$ and adj$(A)$. My attempt: We have \begin{align*} &\psi_{A^{-1}}(x)\\ =&|xI_n-A^{-1}|\\ =&|A^{-1}||xA-I_n|\\ =&|A|^{-1}x^n |A-\frac 1x I_n|\\ =&(-1)^nx^n|A|^{-1}\psi_A(x)\\ =&(-1)^nx^n|A|^{-1} p(x) \end{align*} In this way, we can find the characteristic polynomial for $A^{-1}$ in terms of the characteristic polynomial […]

Sign of det(UV) in SVD

Let $A\in \mathbb{R}^{n\times n}$. Its Singular Value Decomposition (SVD) is $$A=U\Sigma V^T$$ We know $U$ and $V$ are orthogonal matrices. Sometimes $\det UV=1$ and sometimes $\det UV=-1$. My question is: what kind of matrices give $\det UV=1$? Can we say something about the sign of $\det UV$ based on some properties of $A$ before we […]