Articles of linear algebra

If $\lambda$ is an eigenvalue of a nonnegative symmetric matrix with zero diagonal, is $-\lambda$ also an eigenvalue?

If $A$ is an $n \times n$ with $n\geq 2$, non-negative (i.e. no negative elements), symmetric matrix with zero diagonal, is it necessarily the case that the set of all eigenvalues of $A$ can be written $L=\{\pm\lambda_0,\pm\lambda_1,\cdots\}$?

Prove that there are not two matrices 2×2 such that: $AB-BA=I_2$

I tried this question by multiplying explicitly the matrices but I think I’m not getting anything, so I think, well let’s suppose false so $C(AB-BA)=C$ and find a contradiction but also I’m not getting anything.

Prove that $\gcd(pf,pg) = p \cdot \gcd(f,g)$ when $p,f,g$ are polynomials

How does one start proving this theory? Prove: $$\gcd(pf,pg) = p \cdot \gcd(f,g)$$ when $$p,f,g \in \mathbb F[x] \;,\;\text{The max power multiplier of $p$ is 1 (fixed polynomial)}.$$

On the polynomial formula for determinants

I have three questions: 1) For the determinant of a matrix $Q \in \mathbb{R}^{n \times n}$, there is the following polynomial formula: $$ \det(Q)= \sum_\sigma \text{sgn}(\sigma) \prod_{i=1}^{n} Q_{\sigma(i),i} \tag{*} $$ I know about the characteristic polynomial of a matrix whose roots are the eigenvalues, but I am not sure about how (*) comes about? In […]

Do $X(X'X)^{-1}(X'X)^{-1}X'$ and $(X'X)^{-1}$ have the same non-zero eigenvalues?

Let $X$ be a matrix with full column rank. I want to show that $X(X’X)^{-1}(X’X)^{-1}X$ and $(X’X)^{-1}$ have the same non-zero eigenvalues. I have checked it in matlab and Stata and the result holds true for all the examples, but I cannot prove this formally. Please help.

Based on 2 sets of vectors in $\mathbb{R}^4$, how do I determine a system of equations? (Linear Algebra) Please help?

Consider the following 2 sets of vectors in $\mathbb{R}^4$: $A = \{v_1, v_2, v_3\}, B = \{w_1, w_2, v_3\}$. You are given that $A$ is a set of linearly independent vectors and that $B$ is a set of linearly independent vectors. The intersection of 2 sets is the set of elements that are common to […]

Is the image $T(S)$ of a linearly dependent set $S$ under a linear transformation $T$ linearly dependent?

My attempt: $\exists c_1,\cdots,c_k\in F,~\exists v_1,\cdots,v_k\in V,~c_1v_1+\cdots+c_kv_k=0$, where $c_i$ are not all zero. Then $c_1T(v_1)+\cdots+c_kT(v_k)=0$. However, we can not deduce that $T(v_1),\cdots,T(v_k)$ are linearly dependent, since there may be $i,j$ such that $T(v_i)=T(v_j)$, so the coefficients $c_i$ and $c_j$ have the chance to sum up to zero! For example, even though we can get that […]

A numerical scheme for finding radical matrices.

If the goal would be to solve the (matrix) equation: $${\bf P}^2 = {\bf A}$$ Do you think a numerical scheme alternatingly minimizing for $\bf P_1, P_2$ would be stable: $$\|{\bf P_1P_2-A}\|_2,\hspace{1cm} \|{\bf P_1-P_2}\|_2$$ I do not expect there to be any unique solution. On the contrary I am especially interested in the cases when […]

Given two subspaces $N,W$ of $V$ find a linear transformation $T:V\to V$ such that its kernel is $N$ and is range is $W$.

If $N$ and $W$ are subspaces of $V$ such that $\dim(V/N) = \dim W$, then there exists at least one element $A$ of $L(V,V)$ such that $\mbox{ker}(T) = N$ and $\mbox{range}(T) = W$. Two related problem i) Given $N$ subspace of $V$, find a linear transformation $T:V\to V$ with kernel $N$. ii) Given $W$ subspace […]

Get the direction vector passing through the intersection point of two straight lines

Let say I have this diagram, How to find the direction vector passing through the intersection point of two straight lines? Update: new vector is the bisector of two lines and vector may be arbitrary.