A [[Matrix|Matrix]] is [[Symmetric]] if it is equal to its own [[Matrix Transpose|Transpose]]. $\huge A = A^\intercal $ Another definition is a [[Matrix]] $A$ is [[Symmetric Matrix|Symmetric]] [[Bijective|if and only if]] there [[Existential Quantifier|Exists]] an [[Orthogonal]] [[Basis]] of [[Eigenvector|Eigenvectors]] of $A$. ### Properties All [[Eigenvalue|Eigenvalues]] of $A$ are [[Real Numbers|Real Numbers]] (assuming $A$ is composed of [[Real Numbers|Real Numbers]]). If $\vec v_{1},\vec v_{2}$ are [[Eigenvector|Eigenvectors]] of $A$, with respect to [[Eigenvalue|Eigenvalues]] $\lambda_{1} \neq \lambda_{2}$, then $\vec v_{1} \perp \vec v_{2}$. $A$ is [[Eigendecomposition|Diagonalizable]] in the form $PDP^{-1}$, and $D$ will always be [[Symmetric Matrix|Symmetric]] as well. There [[Existential Quantifier|Exists]] an [[Orthogonal Basis|Orthogonal Basis]] of $\R^n$ comprised of [[Eigenvector|Eigenvectors]] of $A$. >[!example] >Let $A \in M_{6 \times 6}$ >$\begin{align} >\det(A-tI) &= >\small >(t-\lambda_{1}) >(t-\lambda_{2}) >(t-\lambda_{3}) >(t-\lambda_{4}) >(t-\lambda_{5}) >(t-\lambda_{6})\\ >\end{align} >$ > >$\begin{align} >\lambda_{2}&=\lambda_3\\ >\lambda_{4} &= \lambda_{5}=\lambda_{6}\\ >\end{align} >$ [[Geometric Multiplicity (LinAlg)|Geometric Multiplicity]] $=$ [[Geometric Multiplicity (LinAlg)|Algabraic Multiplicity]]. $\begin{matrix} \text{\small Eigenvalues}&\lambda_{1} & \lambda_{2},\lambda_{3} & \lambda_{4},\lambda_{5},\lambda_{6} \\ &1 & 2 & 3 \\ \text{\small Eigenspaces}&E_{\lambda_{1}} & E_{\lambda_{2}} & E_{\lambda_{4}} \\ \\ \text{\small Basis} &\set{\vec u_{1}} & \set{\vec u_{2}, \vec u_{3}} & \set{\vec u_{4}, \vec u_{5}, \vec u_{6}} \\ { \small \begin{split} \\ & \text{Orthonormal}\\ &\text{Basis} \end{split} } & \set{\vec w_{1}} & \set{\vec w_{2}, \vec w_{3}} & \set{\vec w_{4}, \vec w_{5}, \vec w_{6}} \\ \end{matrix} $