The Spectral Theorem

Goal

$Q^{T} A Q = D$ for any symmetric matrix $A$

By induction method.
Find $A \sim B$ and $B \sim D ⟶ A \sim D$

ASSUME $Q_{1} = [v_{1} : v_{2} : \dots : v_{n}]$ an orthogonal matrix for which $A v_{1} = λ_{1} v_{1}$ .

\begin{aligned} Q_{1}^{T} A Q_{1} = & [\begin{array}{c} v_{1}^{T} \\ v_{2}^{T} \\ ⋮ \\ v_{n}^{T} \end{array}] A [v_{1} : v_{2} : \dots : v_{n}] \\ = & [\begin{array}{c} v_{1}^{T} \\ v_{2}^{T} \\ ⋮ \\ v_{n}^{T} \end{array}] [A v_{1} : A v_{2} : \dots : A v_{n}] \\ = & [\begin{array}{c} v_{1}^{T} \\ v_{2}^{T} \\ ⋮ \\ v_{n}^{T} \end{array}] [λ_{1} v_{1} : A v_{2} : \dots : A v_{n}] \\ = & [\begin{array}{cc} λ_{1} & * \\ 0 & A_{1} \end{array}] = B \end{aligned}

B^{T} = Q_{1}^{T} A^{T} Q_{1} = Q_{1}^{T} A Q_{1} = B

\Rightarrow B i s s y m m e r t r i c

\Rightarrow A_{1} i s s y m m e t r i c

∴ B = [\begin{array}{cc} λ_{1} & 0 \\ 0 & A_{1} \end{array}]

∵ A \sim B \Rightarrow c_{A} (λ) = c_{B} (λ)

\begin{aligned} ∴ c_{A} (λ) = c_{B} (λ) = & \det (B - λ I) = \det ([\begin{array}{cc} λ_{1} & 0 \\ 0 & A_{1} \end{array}] - λ I) \\ = & \det ([\begin{array}{cc} λ_{1} - λ & 0 \\ 0 & A_{1} - λ I^{'} \end{array}]) \\ = & (λ_{1} - λ) \det (A_{1} - λ I^{'}) \\ = & (λ_{1} - λ) c_{A_{1}} (λ) \end{aligned}

$∴$ The characteristic polynomial of $A_{1}$ divides the characteristic polynomial of $A$ . It follows that the eigenvalues of $A_{1}$ are also eigenvalues of $A$ .

A_{1} i s a k \times k r e a l s y m m e r t r i c m a t r i x \Rightarrow L e t P_{2}^{T} A_{1} P_{2} = D_{1}

Q_{2} = [\begin{array}{cc} 1 & 0 \\ 0 & P_{2} \end{array}] Q = Q_{1} Q_{2}

\begin{aligned} Q^{T} A Q = & (Q_{1} Q_{2})^{T} A (Q_{1} Q_{2}) = (Q_{2}^{T} Q_{1}^{T}) A (Q_{1} Q_{2}) = Q_{2}^{T} B Q_{2} \\ = & [\begin{array}{cc} 1 & 0 \\ 0 & P_{2}^{T} \end{array}] [\begin{array}{cc} λ_{1} & 0 \\ 0 & A_{1} \end{array}] [\begin{array}{cc} 1 & 0 \\ 0 & P_{2} \end{array}] \\ = & [\begin{array}{cc} λ_{1} & 0 \\ 0 & P_{2}^{T} A_{1} P_{2} \end{array}] \\ = & [\begin{array}{cc} λ_{1} & 0 \\ 0 & D_{1} \end{array}] \end{aligned}

The Spectral Theorem ​

The Spectral Theorem