The spectral theorem guarantees, that when we have a symmetric matrix $A$, we can always decompose is into a multiplication of $PDP^T$, where: - $P$ is an [[Orthogonal Matrix]] (square matrix with orthonormal row/column vectors) - $D$ is a [[Matrix Diagonalization#Diagonal Matrix|Diagonal Matrix]]. $ A=PDP^T $ It turns out that the row or column vectors of $P$ are the [[Eigenvectors]] of $A$, and the diagonal elements of $D$ are the eigenvalues of $A$. We can demonstrate this linkage for by multiplying both sides by $v_1$. $ Av_1=(PDP^T)v_1 $ ![[spectral-theorem.jpeg|center|600]] The matrix multiplications lead to the following, which is the definition of eigenvectors and eigenvalues of a matrix $A$. $ Av_1=v_1\lambda_1 $