Consider a $3 imes 3$ real symmetric matrix $S$ such that two of its eigenvalues are $a
eq 0$ and $b
eq 0$ with respective eigenvectors $[x_1\ x_2\ x_3]^T$ and $[y_1\ y_2\ y_3]^T$. If $a
eq b$, then $x_1y_1+x_2y_2+x_3y_3$ equals

Question

Consider a $3	imes 3$ real symmetric matrix $S$ such that two of its eigenvalues are $a
eq 0$ and $b
eq 0$ with respective eigenvectors $[x_1\ x_2\ x_3]^T$ and $[y_1\ y_2\ y_3]^T$. If $a
eq b$, then $x_1y_1+x_2y_2+x_3y_3$ equals

Accepted Answer

0. A real symmetric matrix has orthogonal eigenvectors corresponding to distinct eigenvalues. Since $a
eq b$, the eigenvectors $x$ and $y$ are orthogonal, so their dot product is zero. Hence $x_1y_1+x_2y_2+x_3y_3=0$.

Consider a $3\times 3$ real symmetric matrix $S$ such that two of its eigenvalues are $a\neq 0$ and $b\neq 0$ with respective eigenvectors $[x_1\ x_2\ x_3]^T$ and $[y_1\ y_2\ y_3]^T$. If $a\neq b$, then $x_1y_1+x_2y_2+x_3y_3$ equals

Solution

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