Inner Product Spaces and Forms

Inner products

⟨·,·⟩ must be linear (in one slot), conjugate-symmetric and positive definite. Testing candidates: ${\sum }_j{w}_j{x}_j{y}_j$ is an inner product ⇔ all w_j > 0. Forms like ${\sum }_{i,j}{x}_i{y}_j$ $=(\sum x)(\sum y)$ are degenerate, and $\sum x_j$ $y_{n-j+1}$ is symmetric but indefinite.

Cauchy–Schwarz |⟨x,y⟩| ≤ ‖x‖‖y‖; parallelogram law characterises norms coming from an inner product.

Gram–Schmidt

Produces an orthonormal basis; gives the QR decomposition A = QR. The Gram matrix G = (⟨$v_i,v_j$⟩) is positive semi-definite, and positive definite ⇔ the $v_i$ are independent.

The matrix classes

All the above are normal — normality is exactly the condition for unitary diagonalisability over $\mathbb{C}$.

Spectral theorem consequences

• Real symmetric: eigenvectors for distinct eigenvalues are orthogonal; $A=Q\Lambda Q^T$.
• A symmetric with $A^k=I\Rightarrow A^2=I($real eigenvalues with ${\lambda }^k=1\Rightarrow \lambda =\pm 1)$.
• A symmetric with Aᵏ = 0 ⇒ A = 0.
• Positive definite ⇔ all eigenvalues > 0 ⇔ all leading principal minors > 0 ⇔ A = BᵀB with B invertible.

Orthogonal projections

P is an orthogonal projection $\iff P^2=P$ and P = Pᵀ. Then ${\mathbb{R}}^n=$ im P ⊕ ker P orthogonally, and rank P = trace P.

Key takeaways

• Normal ⇔ unitarily diagonalisable; symmetric/Hermitian/unitary/skew are all special cases.
• Real symmetric gives a real orthonormal eigenbasis — the strongest statement.
• Positive weights are what make $\sum w_j{x}_j{y}_j$ an inner product.