Question about the standard orthonormal basis of nomalized eigenvectors of a positive definite matrix

Question

Given a symmetric positive definite matrix $A$. May someone please explain the following statement for me?

Let $v_1, ..., v_n$ be the standard orthonormal basis of $A$. We define a vector, that is not orthogonal to $v_1$ by $\sum_{i=1}^n c_iv_i$ Hence, we obtain $\sqrt{\sum_{i=1}^n c_i^2}=1$. Unfortunately in my lecture there were no details on $c_i$. Does somebody know why this is true?

Answer 1

What is a basis of a matrix ??

Let $w=\sum_{i=1}^n c_iv_i.$ Let us denote the usual inner product on $ \mathbb R^n$ (or $ \mathbb C^n$) by $(\cdot, \cdot)$ and the induced norm $|| \cdot||= \sqrt{(\cdot, \cdot)}.$

By Pythagoras we get

$$||w||^2= \sum_{i=1}^n||c_i v_i||^2= \sum_{i=1}^nc_i^2.$$

If $||w||=1$, then we have $\sum_{i=1}^nc_i^2=1$.

Do we have $||w||=1 ?$