Articles of linear algebra

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 […]

Coefficient Matrix and Properties of $\mathcal{L}_{B}: V\rightarrow \mathbb{R}^{3}$

Having a bit of trouble with the following question setup: Denote by $\mathbb{R}[x,y]$ the set of polynomials with two variables $x$ and $y$ and real coefficients. Note that $\mathbb{R}[x,y]$ forms a linear space under polynomial addition and scalar multiplication. Consider a set of elements $\mathcal{B}=(x^2,xy,y^2).$ Denote by $V=Span(\mathcal{B})$ the linear sub-space spanned by $\mathcal{B}$. Define […]

Factoring a polynomial (multivariable)

Factor $ (a – b)^3 + (b – c)^3 + (c-a)^3$ by SYMMETRY. Okay, this is the problem. Let $f(a) = (a – b)^3 + (b-c)^3 + (c-a)^3$ obviously, if you let $a = b$ then, $f(b) = 0$, thus $(a – b)$ is a factor of $f(a)$. Then someone said : If $(a – […]

Determinant of determinant is determinant?

Looking at this question, I am thinking to consider the map $R\to M_n(R)$ where $R$ is a ring, sending $r\in R$ to $rI_n\in M_n(R).$ Then this induces a map. $$f:M_n(R)\rightarrow M_n(M_n(R))$$ Then we consider another map $g:M_n(M_n(R))\rightarrow M_{n^2}(R)$ sending, e.g. $$\begin{pmatrix} \begin{pmatrix}1&0\\0&1\end{pmatrix}&\begin{pmatrix}2&1\\3&0\end{pmatrix}\\ \begin{pmatrix} 0&0\\0&0 \end{pmatrix}&\begin{pmatrix} 2&3\\5&2\end{pmatrix} \end{pmatrix}$$ to $$\begin{pmatrix}1&0&2&1\\ 0&1&3&0\\ 0&0&2&3\\ 0&0&5&2\end{pmatrix}.$$ Is it true […]

Find the implicite and explicite set of this little cartesian product

$S$ is the set of all $(x_{1},x_{2}) \in \mathbb{R} \times \mathbb{R}$ with $x_{1}+x_{2} \geq 3$ and $-x_{1}+2x_{2}=6$ What’s its implicit and explicit set? Implicite: $S=\left\{(x_{1},x_{2})|x_{1},x_{2} \in \mathbb{R}\right\}$ For the explicite we somehow have to calculate the solutions for $x_{1}$ and $x_{2}$ (?) The problem is there is this inequality sign… $$-x_{1}+2x_{2}=6$$ $$x_{1}=2x_{2}-6$$ Now take the […]

Exterior Algebra: General Case, find a linear operator

Is true that for all $J:\wedge^{k} \mathbb{R}^{n} \to\wedge^{k} \mathbb{R}^{n}$ isomorphism linear there exists $A:\mathbb{R}^{n} \to \mathbb{R}^{n}$ linear operator such that $\wedge^{k} A =J$? When $k=(n-1)$, we have a positive answer, just do like this. Related: Exterior Algebra: Find a linear operator