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- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/07%3A_Inner_Product_Spaces/7.03%3A_Orthogonal_DiagonalizationIf A is an n×n symmetric matrix, then TA:Rn→Rn is a symmetric operator, so let B be an orthonormal basis of Rn consisting of eige...If A is an n×n symmetric matrix, then TA:Rn→Rn is a symmetric operator, so let B be an orthonormal basis of Rn consisting of eigenvectors of TA (and hence of A ). Using the inner product ⟨a+bx+cx2,a′+b′x+c′x2⟩=aa′+bb′+cc′, show that T is symmetric and find an orthonormal basis of P2 consisting of eigenvectors.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_TheoryThis page discusses Eigenvalues and Eigenvectors in Spectral Theory, covering special matrices, diagonalization, applications, Markov matrices, dynamical systems, orthogonal diagonalization, singular ...This page discusses Eigenvalues and Eigenvectors in Spectral Theory, covering special matrices, diagonalization, applications, Markov matrices, dynamical systems, orthogonal diagonalization, singular value decomposition, special factorizations, and quadratic forms. It includes exercises for practice to enhance understanding of both theoretical and practical aspects of these concepts.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/08%3A_Orthogonality/8.02%3A_Orthogonal_Diagonalization\[\begin{aligned} (P_{1}P_{2})^TA(P_{1}P_{2}) &= P_{2}^T(P_{1}^TAP_{1})P_{2} \\ &= \left[ 100QT\right] \left[ \begin{array}{cc} \lambda_{1} & 0 \\ 0 & A_{1} \e...(P1P2)TA(P1P2)=PT2(PT1AP1)P2=[100QT][λ100A1][100Q]=[λ100D1]
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/07%3A_Orthogonality/7.06%3A_Orthogonal_Diagonalization\[\begin{aligned} (P_{1}P_{2})^TA(P_{1}P_{2}) &= P_{2}^T(P_{1}^TAP_{1})P_{2} \\ &= \left[ 100QT\right] \left[ \begin{array}{cc} \lambda_{1} & 0 \\ 0 & A_{1} \e...(P1P2)TA(P1P2)=PT2(PT1AP1)P2=[100QT][λ100A1][100Q]=[λ100D1]
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/10%3A_Inner_Product_Spaces/10.03%3A_Orthogonal_DiagonalizationIf A is an n×n symmetric matrix, then TA:Rn→Rn is a symmetric operator, so let B be an orthonormal basis of Rn consisting of eige...If A is an n×n symmetric matrix, then TA:Rn→Rn is a symmetric operator, so let B be an orthonormal basis of Rn consisting of eigenvectors of TA (and hence of A ). Using the inner product ⟨a+bx+cx2,a′+b′x+c′x2⟩=aa′+bb′+cc′, show that T is symmetric and find an orthonormal basis of P2 consisting of eigenvectors.
