Linear Algebra: Quadratic Forms, Eigenvalues, and Matrix Decomposition

Quadratic Forms and Symmetric Matrices

The quadratic form is defined as w(x) = x^TAx, where M(w)_c = 1/2(A + A^T) is the associated symmetric matrix in the canonical basis.

Change of basis formula: M(w)_b = (P_BC)^T · M(w)_c · P_BC.

Rank properties: rg(w) = rg(A) = rg(M(w)_c). If rg(w) = max, the form is non-degenerate; otherwise, it is degenerate.

Diagonalization of Symmetric Matrices

For a symmetric matrix A, applying elementary row and column operations leads to (D | P^T), where D is diagonal. The goal is to obtain zeros in the off-diagonal positions.

A form w admits a non-degenerate orthonormal basis if the rank is maximum or the diagonal elements of D are non-zero.

Signature and Sylvester's Criterion

The signature sig(w) = sig(A) = (p, r_p), where rg(w) = p + r_p. Sylvester's criterion uses principal minors A_k: if A_k > 0, the matrix is positive definite; if (-1)^kA_k > 0, it is negative definite.

Orthonormalization and QR Decomposition

Gram-Schmidt Process

• u₁ = e₁ / |e₁|
• u'₂ = e₂ - <e₂, u₁> u₁
• u₂ = u'₂ / |u'₂|
• u'₃ = e₃ - <e₃, u₁> u₁ - <e₃, u₂> u₂

QR Factorization

For A ∈ M_mxn with rg(A) = k, there exists Q (with orthonormal columns) and R (upper triangular with positive diagonal elements) such that A = QR. Here, rank(A) = rank(R) = rank(Q) = k.

Eigenvalues and Diagonalization

The characteristic equation is p(λ) = det(A - λI) = 0. Eigenvectors generate subspaces for each eigenvalue.

Diagonalization Conditions

A matrix is diagonalizable if P^-1AP = D. This requires that for each eigenvalue, the algebraic multiplicity (ma) equals the geometric multiplicity (mg), where mg(λ) = dim(ker(A - λI)) = n - rank(A - λI).

Matrix Properties

• The sum of eigenvalues equals the trace of A.
• The product of eigenvalues equals the determinant of A.
• Similar matrices A and B satisfy P^-1AP = B.

Matrix Exponentials and Trigonometry

Key identities:

• det(e^A) = e^tr(A)
• e^mA = [e^A]^m
• For A = [[a, b], [-b, a]], e^At = e^at [[cos(bt), sin(bt)], [-sin(bt), cos(bt)]]
• cos(A) = (e^iA + e^-iA) / 2
• sin(A) = (e^iA - e^-iA) / 2i