Let P be a real square matrix satisfying $P = P^T$ and $P^2 = P$. i) Can the matrix P have complex eigenvalues?

Question

Let P be a real square matrix satisfying $P = P^T$ and $P^2 = P$ .
i) Can the matrix P have complex eigenvalues? If so, construct an example, else, justify your answer.
ii) What are the eigenvalues of P?

Solution

Part-1: Conceptualize the Matrix P

$P=P^T$ implies that P is a Symmetric Matrix. This means that except the diagonal elements, all other elements of the matrix are like $a_{ij}=a_{ji}$ .

Thus,

$P = \begin{bmatrix} a & c \\ c & b \end{bmatrix}$

Let’s consider this sample. Assuming that whatever non-size related property that holds true for this 2 x 2 matrix should hold true for an n x n matrix as well.

Part-2: Construct the Matrix P

We know that, $P^2=P$

Thus,

$P = \begin{bmatrix} a & c \\ c & b \end{bmatrix}.\begin{bmatrix} a & c \\ c & b \end{bmatrix} = \begin{bmatrix} a & c \\ c & b \end{bmatrix}$

$\Rightarrow \begin{bmatrix} (a^2 + c^2) & c.(a + b) \\ c.(a + b) & (b^2 + c^2) \end{bmatrix} = \begin{bmatrix} a & c \\ c & b \end{bmatrix}$

(1) $\begin{equation*} \left\ a^2 + c^2 = a \end{equation*}$

(2) $\begin{equation*} \left\ b^2 + c^2 = b \end{equation*}$

(3) $\begin{equation*} \left\ c.(a + b) = c \end{equation*}$

(4) $\begin{equation*} \left\ (1) \ and \ (2) \ holds \ if \ a=b=1 \end{equation*}$

(4) & (3) $\Rightarrow c=0$

Thus, matrix P is an Identity matrix.

$P = I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Part-3: Find Eigenvalues of P

$\begin{align*} \left Px = \lambda x \Rightarrow (P - \lambda I)x=0 \end{align*}$

$\begin{align*} \left P - \lambda I = 0 \Rightarrow I - \lambda I=0 \Rightarrow I(1 - \lambda)=0 \end{align*}$

Thus, Eigenvalue of P is λ=1

Part-4: Can the matrix P have complex eigenvalues?

From Part 3, we see that P cannot have complex values as eigenvalues.

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