= Orthonormal eigenbasis
{title2=$\langle e_i,e_j\rangle=\delta_{ij},\quad Ae_j=\lambda_je_j$}
= Orthonormal eigenbases
{synonym}
An orthonormal eigenbasis of a <linear operator> is an <orthonormal basis> consisting of <eigenvectors>. Each <vector> can be reconstructed from its scalar components along those mutually perpendicular unit directions. With the convention that the <inner product> is linear in its first argument,
$$
x=\sum_j\langle x,e_j\rangle e_j,\qquad
Ax=\sum_j\lambda_j\langle x,e_j\rangle e_j.
$$
The <finite-dimensional spectral theorem> supplies such a basis for a <self-adjoint operator>. In its two-dimensional example, the <coordinates> of $x=(1,0)^T$ are $1/\sqrt2$ along $e_+$ and $-1/\sqrt2$ along $e_-$. The operator multiplies those coordinates by $3$ and $1$, and reconstruction gives $Ax=(2,1)^T$.
Repeated <eigenvalues> permit different orthonormal choices within one <eigenspace>, so the basis need not be unique. The <spectral projection of a self-adjoint matrix> onto that entire <eigenspace> is unique: it keeps the geometrically meaningful component without choosing individual axes inside it.
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