I was wondering if anyone could help me with this problem: Find all a's such that the matrix is diagonalizable: $\begin{bmatrix} 1 & 1\\ a & 1 \end{bmatrix}$
I have to following Lie Algebra $L=\{x\in End(\mathbb{C}^6)\colon x^tS+Sx=0\}$, where $S=[\begin{smallmatrix} 0&I_3 \\ I_3&0 \end{smallmatrix}]$, and the subalgebra $H$ given by the diagonal elements in $L$.
I have found a basis of $L$ and $H$, which are, respectively:
$B_L=\{e_{ij}-e_{i+3j+3},...
The question:
Let $S = \left(\begin{array}{cc} 0 & I_3 \\ I_3 & 0 \end{array}\right)$ and let $$L = \mathfrak{so}_6(\mathbb{C})= \{x \in \operatorname{End}(\mathbb{C}^6)\ |\ ^{\mathrm t}xS + Sx = 0 \}$$
1) Find a basis for $L$ and prove that $\dim L =15$ (Hint: write $x = \left(\begin{array}{c...
I have to following Lie Algebra $L=\{x\in End(\mathbb{C}^6)\colon x^tS+Sx=0\}$, where $S=[\begin{smallmatrix} 0&I_3 \\ I_3&0 \end{smallmatrix}]$, and the subalgebra $H$ given by the diagonal elements in $L$.
I have found a basis of $L$ and $H$, which are, respectively:
$B_L=\{e_{ij}-e_{i+3j+3},...
Linear & Abstract algebra
Linear & Abstract algebra