Home | | Latest | About | Random
# Some preliminary collection of seemingly random facts.
Show or verify the following facts.
![[---images/---assets/---icons/question-icon.svg]] Show every complex number $a+ib$ has a square root. In particular $$
\pm \left( \sqrt{\frac{a+\sqrt{a^2+b^2}}{2}} + i \sqrt{\frac{-a+\sqrt{a^2+b^2}}{2}} \right)
$$are square roots of $a+ib$.
![[---images/---assets/---icons/question-icon.svg]] Show aa a corollary, every complex monic quadratic polynomial $t^2+\lambda t+\mu$ has two complex roots, for any $\lambda,\mu\in\mathbb{C}$.
![[---images/---assets/---icons/question-icon.svg]] Show every odd degree real polynomial has a real root.
![[---images/---assets/---icons/question-icon.svg]] Suppose $n\times n$ matrices $X,Y$ over field $\mathbb{F}$ commutes, and $\lambda$ is an eigenvalue of $X$. Denote $K=\ker(X-\lambda I)$ be the corresponding eigenspace and $W=\operatorname{im}(X-\lambda I)$ be the image of $X-\lambda I$. Show $Y$ is stable on both $K$ and $W$, namely $YK\subset K$, and $YW\subset W$. That is to say, we can consider $Y$ as a linear operator restricted on $K$ as well as a linear operator restricted on $W$.
![[---images/---assets/---icons/question-icon.svg]] Let $p(t)=t^n + a_{n-1} t^{n-1}+\cdots+a_0$ be any monic polynomial over field $\mathbb{F}$. Then the $n\times n$ matrix $$
C_p= \begin{pmatrix}
0 & 0 & \cdots & 0& -a_0\\
1 & 0 & & 0& -a_1\\
0 & 1 & & 0 & -a_2\\
\vdots & & \ddots & 0 & \vdots\\
0 & & 0 & 1 & -a_{n-1}
\end{pmatrix}
$$called its associated **companion matrix** has characteristic polynomial $p$.
![[---images/---assets/---icons/question-icon.svg]] Any eigenvalue to square matrix $A$ is a root to its characteristic polynomial.
![[---images/---assets/---icons/question-icon.svg]] Denote $H_n=\operatorname{Herm}_n(\mathbb{C})$ to be the set of $n\times n$ complex matrices that are Hermitian (namely, self-adjoint). that is $A\in H_n$ if and only if $A^\ast = (\bar A)^T$. It consists of matrices equal to its own complex conjugate transpose. Show $H_n$ is a real vectorspace with dimension $n^2$ over $\mathbb{R}$. Can $H_n$ be considered as a vectorspace over $\mathbb{C}$?
![[---images/---assets/---icons/question-icon.svg]] Show if $M\in \operatorname{Herm}_n(\mathbb{C})$, then every eigenvalue of $M$ is real.
![[---images/---assets/---icons/question-icon.svg]] Fix any $n\times n$ square matrix $A$ over $\mathbb{C}$, it can be any matrix. Show the two linear maps $L_1,L_2:H_n\to H_n$ are well-defined **and** they commute, $L_1L_2 = L_2 L_1$, where $$
\begin{align*}
L_1(X) & = \frac{1}{2}(AX+XA^\ast) \\
L_2(X) & = \frac{1}{2i}(AX-XA^\ast)
\end{align*}
$$
![[---images/---assets/---icons/question-icon.svg]] Denote $\operatorname{Sk}_n= \operatorname{Skew}_n(\mathbb{C})$ to be the space of all $n\times n$ complex skew-symmetric matrices, where $A\in \operatorname{Sk}_n$ is such that $A^T=-A$. Show $\dim_{\mathbb{C}}(\operatorname{Sk}_n)=\frac{n(n-1)}{2}$, as a vectorspace over $\mathbb{C}$.
![[---images/---assets/---icons/question-icon.svg]] Take any complex $n\times n$ matrix $A$. Define $L_1,L_2 : \operatorname{Sk}_n\to \operatorname{Sk}_n$ where $$
\begin{align*}
L_1 (X) & =AX-XA^T \\
L_2(X) & =AXA^T
\end{align*}
$$Show $L_1$ and $L_2$ are well-defined, and that they commute, $L_1L_2=L_2L_1$.
![[---images/---assets/---icons/question-icon.svg]] Show if $n > 1$ is a positive integer such that $2^\ell \not\mid n$ but $2^{\ell-1} \mid n$, then $2^{\ell-1} \not\mid \frac{n(n-1)}{2}$, where $\ell > 1$.