Let $V$ a complex vector space with dimension n and inner product ,$u \in V $ unit vector. Let $ H_u: V \rightarrow V$ defined by $ H_u(v) = v - 2 <v,u>u$ $\forall v \in V$. Then:
a)$H_u(u) = -u$
b)$H_u(v) = v$ iff $v$ is orthogonal to $u$
c) Find the characteristic polynomial of $H_u$
d) Find the spectral decomposition of $H_u$
I could prove a) and b) but how can i solve c) and d) some help please. I don't understand the concept of spectral decomposition some clarity please?
Now we can make an orthonormal basis of eigenvectors of $H_u$, by extending $u$ by an arbitrary orthonormal basis $v_1,\dots,v_{n-1}$ of the $n-1$ dimensional subspace $u^\perp$.
Can you write up the matrix of $H_u$ in this basis $(u,v_1,\dots,v_{n-1})$?
The characteristic polynomial of a linear transformation that has a basis of eigenvectors, is $\prod_{i=1}^n(x-\lambda_i)$ where $\lambda_i$ are the eigenvalues.
Now $u$ is an eigenvector with eigenvalue $-1$ (as $H_u(u)=(-1)\cdot u$), and each $v_j$ has eigenvalue $1$.