Articles of linear algebra

preservation of extreme points under linear transformation

Suppose $\{e_1,…,e_N\}$ is the set of all extreme points of a compact convex subset $X\subset\mathbb R^n$. $L: \mathbb R^n\to \mathbb R^m$ is a linear transformation. $L$ is surjective but is not injective. Let $Y= L(X)$. Would it hold that for every $1\leq i\leq N$, $L(e_i)$ must be an extreme point of $Y$? Is there any […]

Prove that $A$ and $B$ are invertible and $B=A^{-1}$

Suppose $A$ and $B$ are $n \times n$ matrices. Assume $AB=I$. Prove that $A$ and $B$ are invertible and that $B=A^{-1}$. Please let me know whether my proof is correct and if there are any improvements to be made. Assume $AB=I$. Then $(AB)A=IA=A$. So, $A(BA)=AI=A$. Then $BA=I$. Therefore $AB=BA=I$. Thus $A$ and $B$ are invertible. […]

$\sum\limits_{j,k} {{a_{jk}}} {\bar u_j}{u_k}\mathop = \limits^? \int_0^1 | \sum\limits_j {{u_j}} {x^j}{|^2}x{\mkern 1mu} dx$

Let $A\in M_n$ and $U\in \mathbb{C}^n$ and $a_{jk}=\int_0^1 x^{j+k-1}\,dx$ Is this true that $\sum_{j,k} a_{jk}\bar u_ju_k=\int_0^1\Bigl|\sum_j u_jx^j\Bigr|^2x\,dx$

Linear transformation of a subspace

How can I calculate the image of a linear transformation of a subspace? Example: Given a subspace $A$ defined by $x + y + 2z=0$, and a linear transformation defined by the matrix $$M= \left( \begin{matrix} 1 & 2 & -1\\ 0 & 2 & 3\\ 1 & -1 & 1\\ \end{matrix}\right) $$ What is […]

Exponential of a matrix and related derivative

I have $ X \in M(n,\mathbb R) $ to be fixed. I define $ g(t) = \det(e^{tX}) $ Then the author proceeds as follows: $ g'(s) = \frac {d}{dt} g(s+t) $ = $ \frac {d}{dt} \det(e^{(s+t)X}) |_{t=0} $ = $ \frac{d}{dt}(\det(e^{sX})\det(e^{tX})|_{t=0} $ = $ g(s)\tr(X) $, as $ s $ is independent of $ t […]

show invertibility of linear transformations

This question already has an answer here: Showing a Linear Transformation is invertible 2 answers

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix $A$? Thank you very much!

Ratio of area formed by transformed and original sides of a parallelogram

I am interested in finding the ratio of area formed by transformed and original sides of a parallelogram, given by: $$\frac{\|Ma\times Mb\| }{\| a\times b \|}$$ $M$ is a $3 \times 3$ matrix and $ a, b$ are vectors with 3 components each ($a,b$ are sides of the original parallelogram and $Ma, Mb$ are sides […]

Non-linear matrix equation solvable with linear algebra?

Consider the matrix equation $${\bf X}^k\bf {A = B}$$ Which we want to solve for $\bf X$ We can put A and B in a “block-vector”: $v = [{\bf A}^T,{\bf 0},\cdots,{\bf 0},{\bf B}^T]^T$, assume there exists a matrix: $$M = \begin{bmatrix}\bf I&\bf 0&\bf 0&\cdots&\bf 0\\\bf X_1&\bf 0&\bf 0&\cdots&\bf 0\\\bf 0&\bf X_2&\bf 0&\cdots&\bf 0\\\vdots&\ddots&\ddots&\cdots&\bf 0\\\bf 0&\bf […]

power series for square root matrix

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with entries following $|X_{ij}|\le1$. If $z$ has a small modulus ($|z|\ll1$), am I allowed to expand in a power series the square root matrix expression […]