Hello world and welcome to our new website!

We have decided to move away from Wix and migrate to wordpress. This offers many advantages to us.

Notably we now have math support, [katex]x^2[/katex].

#invariants

The dot product of a vector with itself is always non-negative, [katex]\mathbf{v} \cdot \mathbf{v} \geq 0[/katex]

The determinant of an orthogonal matrix is always 1, [katex]\det(\mathbf{O}) = 1, \text{where } \mathbf{O} \mathbf{O}^T = \mathbf{I}[/katex]

The trace is invariant under cyclical permutations, [katex]\text{Tr}(\mathbf{A} \mathbf{B} \mathbf{C}) = \text{Tr}(\mathbf{C} \mathbf{A} \mathbf{B})[/katex]

The trace of the matrix is invariant under changes of basis, [katex]\lambda_1 + \lambda_2 + \dots + \lambda_n = \text{Tr}(\mathbf{A})[/katex]

The sum of principle minors is invariant, [katex] \text{det} \begin{pmatrix}
a_{1,1} & & & \\
a_{2,1} & a_{2,2} & & \\
\vdots & & \ddots & \\
a_{n,1} & a_{n,2} & \dots & a_{n,n}
\end{pmatrix} = \sum_{i=1}^n (-1)^{i+j} a_{i,j} \text{det} \begin{pmatrix}
a_{1,1} & a_{1,2} & \dots & a_{1,i-1} & a_{1,i+1} & \dots & a_{1,n} \\
a_{2,1} & a_{2,2} & \dots & a_{2,i-1} & a_{2,i+1} & \dots & a_{2,n} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
a_{i-1,1} & a_{i-1,2} & \dots & a_{i-1,i-1} & a_{i-1,i+1} & \dots & a_{i-1,n} \\
a_{i+1,1} & a_{i+1,2} & \dots & a_{i+1,i-1} & a_{i+1,i+1} & \dots & a_{i+1,n} \\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
a_{n,1} & a_{n,2} & \dots & a_{n,i-1} & a_{n,i+1} & \dots & a_{n,n}
\end{pmatrix}[/katex]