# The set St(k, n), k < n, is called the Stiefel set.

The set St(k, n), k < n, is called the Stiefel set.

does matrix X= $$\left[ \begin{array}{ccc} 0&-{\sqrt 2}/2 \\ 0&{\sqrt 2}/2\\ 1&0 \end{array} \right]$$

belong to St(2,3)?

Im not really sure how to solve this. I dont understand what a stiefel set is and i've tried reading about it, but i still dont get it.

According to Wikipedia, and Wolfram Mathworld, a Stiefel manifold is a set of ordered $$\ k$$-tuples, $$\ \left(v_1, v_2, \dots, v_k\right)\$$, of orthonormal vectors in $$\ \mathbb{R}^n\$$. Presumably, this is what your expression $$\ \text{St}(k,n)\$$ is intended to represent.

So if you treat your matrix $$\ X\$$ as an ordered set of columns, then it will lie in $$\ \text{St}(2,3)\$$ if and only it's an ordered set of $$\ 2\$$ orthonormal columns in $$\ \mathbb{R}^3\$$. Can you see whether or not that's true?

• could you explain more please? – ELZP Feb 21 at 8:03