Chapter 4.CE, Problem 14CE

In Exercises $9-14$, give a brief answer.

Let u be a vector in $R^n$ such that $u^{T}u=1$, and let A denote the $\left(n\times n\right)$ matrix $A=I-2u{u}^T$

a) Is A is symmetric?

b) Is A is orthogonal?

c) Calculate $\mathrm{A}\mathbf{u}$

d) Suppose that $\mathbf{w}$ is in $R^n$ and $u^{T}w=0$. What is Aw?

e) Give the eigenvector of A and give the geometric multiplicity for each eigenvlaue