Articles of linear algebra

Dimension of a vector space is even implies Ker T = Im T.

Let $ V $ be a vector space space of dimension $ n $ even, then there exists a linear map $T : V \rightarrow V $ such that $ Ker T = Im T $ ? The converse is obviously true.

Orthogonal basis for this indefinite symmetric bilinear form

Find an orthogonal basis for the bilinear form over $\mathbb{R}$ given by $(\mathbf{x}, \mathbf{y})\mapsto\mathbf{x}^{t}A\mathbf{y}$ where $A=\begin{bmatrix} 1 & 4 & 4\\ 4 & 4 & 10\\ 4 & 10 & 16 \end{bmatrix}$. I’m not sure if this is as easy as using Gram-Schmidt, or if there is another way. I used Gram-Schmidt and obtained very […]

How to solve simple systems of differential equations

Say we are given a system of differential equations $$ \left[ \begin{array}{c} x’ \\ y’ \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix} $$ Where $A$ is a $2\times 2$ matrix. How can I in general solve the system, and secondly sketch a solution $\left(x(t), y(t) \right)$, in the $(x,y)$-plane? For example, let’s say $$\left[ […]

Hoffman and Kunze, Linear algebra Sec 2.4 Theorem 8

Theorem 8. Suppose $P$ is an $n\times n$ invertible matrix over $F.$ Let $V$ be an $n$-dimensional vector space over $F,$ and let $\scr B$ be an ordered basis of $V.$ Then there is a unique ordered basis $\scr \overline{B}$ of $V$ such that \begin{align} \tag{1}[\alpha]_{\scr B} = P[\alpha]_{\scr \overline{B}} \end{align} for every vector $\alpha […]

A theorem of symmetric positive definite matrix.

Is the following true? Let $g=(g_{ij})\in M(n,\Bbb R)$ be a symmetric positive-definite matrix and let $a=(a_1,\ldots,a_n)\in\Bbb R^n$ be any vector. Then, $$v^Tgv=1\implies (v\cdot a)^2\leq a^Tg^{-1}a,$$ or in other words, $$g_{ij}v^iv^j=1\implies v^iv^ja_ia_j\le g^{ij}a_ia_j,$$ (with Einstein summation). I was trying to prove that the gradient $$grad(f)=g_{ij}\frac{\partial f}{\partial x^i}\frac{\partial }{\partial x^j}$$ on a Riemannian manifold $(M,g)$ is the […]

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 […]

In a $p$-adic vector space, closest point on (and distance from) a plane to a given point?

Let $\| x \| =\sqrt{x^T x}$ be the Euclidean norm on $\mathbb{R}^n$. Consider the point $z \in \mathbb{R}^n$, and the plane $P = \{x \in \mathbb{R}^n : a^T x = b\}$ where $0 \neq a \in \mathbb{R}^n$, $b \in \mathbb{R}$. Orthogonal projection gives the point \begin{align}\label{1}\tag{1} y = z – \frac{(a^T z – b)}{a^T a} […]

How similarity transformation is related to coordinate transformation?

I know that every matrix can be transformed into its Jordan form using similarity transformation. But I wanted to know, this transformation is related to shifting of coordinate systems?

Should I use sets or tuples when dealing with linear dependence?

Let set of vectors $\{x,y,z\}$ be linearly independent. Then would $\{x,y,z,x\}=\{x,y,z\}$ be linearly dependent, also? If so, that seems like a problem (since $\alpha x+\beta y+\gamma z+(-\alpha)x=0$ would allow for a non-zero $\alpha$) unless it is understood that each vector in a set of vectors is used only once in a linear combination of that […]

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 […]