Computing explicit Riesz projection

Question

Consider the matrix: $ A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $ which has eigenvalue $1$ with algebraic multiplicity $2$ and geometric multiplicity $1$. I am trying to explictly construct the Riesz projection \begin{align*} P = - \frac{1}{2 \pi i} \int_{\gamma} \frac{1}{A - z} d \gamma \end{align*}  where $\gamma$ is any curve enclosing the spectrum $\{1\}$. How to compute it?

Answer 1

It seems easiest to pick the contour $\gamma: \lbrack 0, 2 \pi \rbrack \to \mathbb{C} $ defined by \begin{align*} \gamma( \theta) = 1+ e^{i \theta} \end{align*}  which is the circle of unit radius around $1$ and $\gamma'(\theta) = i e^{i \theta} $. Now, to evaluate the contour integral \begin{align*} - \frac{1}{2 \pi i} \int_{0}^{2\pi} \frac{i e^{i \theta}}{ \begin{pmatrix} - e^{i \theta} & 1 \\ 0 & - e^{i \theta} \end{pmatrix}} d \theta & = - \frac{1}{2 \pi i} \int_{0}^{2\pi} \frac{i e^{i \theta}}{e^{2i\theta}}   \begin{pmatrix} - e^{i \theta} & - 1 \\ 0 & - e^{i \theta} \end{pmatrix} d \theta = \frac{1}{2 \pi } \int_{0}^{2\pi}   \begin{pmatrix}  1 & e^{-i \theta} \\ 0 & 1 \end{pmatrix} d \theta \\ & = \frac{1}{2 \pi } \begin{pmatrix} \int_{0}^{2\pi}   1 d\theta & \int_{0}^{2\pi}   e^{-i \theta} d \theta \\ 0 & \int_{0}^{2\pi}   1 d \theta \end{pmatrix} =   \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \end{align*} Which is a projection onto the entire $\mathbb{C}^2$ as expected.