Let A=I_2-2MM^T, where M is a real matrix of order 21 such that M^TM=I_1.

Question

Let $A=I_2-2MM^T$, where $M$ is a real matrix of order $2	imes1$ such that $M^TM=I_1$. If $\lambda$ is a real number such that $AX=\lambda X$ holds for some non-zero real matrix $X$ of order $2	imes1$, then the sum of squares of all possible values of $\lambda$ is equal to:

Answer: 2

Accepted Answer

Since $M^TM=1$ (a scalar), $A=I-2MM^T$ is a Householder reflection matrix: it satisfies $A^T=A$ and $A^2=I-4MM^T+4M(M^TM)M^T=I-4MM^T+4MM^T=I$, so $A^2=I$.

Hence every eigenvalue $\lambda$ of $A$ satisfies $\lambda^2=1$, i.e. $\lambda=\pm1$. Since $A
e\pm I$ in general (it's a genuine reflection, $M
e0$), both eigenvalues $1$ and $-1$ occur.

Sum of squares of all possible $\lambda$: $1^2+(-1)^2=2$.

Solutions · 1