设n阶矩阵A有n个两两正交的特征向量，证明A是对称矩阵。

<br> 设A的n个两两正交的特征向量为<ruby>α<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>₁,<ruby>α<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>₂,…,<ruby>α<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>_n,其对应的特征值依次为λ₁,λ₂,…,λ_n。<br> 令<ruby>ξ<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>_i=<ruby>α<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>_i/,<ruby>α<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>_i,(i=1,2,…,n),则<ruby>ξ<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>₁,<ruby>ξ<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>₂,…,<ruby>ξ<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>_n是两两正交的单位向量。<br> 记P=(<ruby>ξ<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>₁,<ruby>ξ<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>₂,…,<ruby>ξ<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">→</rt><rp>)</rp></ruby>_n),即P是正交矩阵。从而有P^-¹=P^T,P^-¹AP=diag(λ₁,λ₂,…,λ_n)=Λ,即A=PΛP^-¹=PΛP^T,故A^T=(PΛP^T)^T=(P^T)^TΛ^TP^T=PΛP^T=A,即A是对称矩阵。