Question

Let $A = I_2 - MM^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M = I_1$ holds. If $\lambda$ is a real number such that the relation $AX = \lambda X$ holds for some non-zero real matrix $X$ of order $2 \times 1$, then the sum of squares of all possible values of $\lambda$ is equal to:

Answer 1

We know that $A = I_2 - MM^T$ and $M^TM = I_1$, which implies that $M$ is a unit vector. The matrix $A$ is a projection matrix, and for a projection matrix, the eigenvalues are either 0 or 1.

In this case, the eigenvalue $\lambda$ of $A$ can be either 0 or 1. Therefore, the sum of squares of all possible values of $\lambda$ is:

$0^2 + 1^2 = 1 + 1 = 2.$

Answer 2

We know that $A = I_2 - MM^T$ and $M^TM = I_1$, which implies that $M$ is a unit vector. The matrix $A$ is a projection matrix, and for a projection matrix, the eigenvalues are either 0 or 1.

In this case, the eigenvalue $\lambda$ of $A$ can be either 0 or 1. Therefore, the sum of squares of all possible values of $\lambda$ is:

$0^2 + 1^2 = 1 + 1 = 2.$