LINEAR ALGEBRA 2: CHEAT SHEET
BY SHERWIN NAIDOO
1. Eigenvalues and Eigenvectors
• Eigenvalue equation: Ax = λx
• Characteristic Polynomial: 𝑑𝑒𝑡(A − 𝜆I) = 0
• Eigenvalues (λ): Solutions to the characteristic polynomial.
• Eigenvectors (x): refers to non-zero vectors satisfying, Ax = 𝜆x.
2. Diagonalization:
• Diagonalizable matrix: A = PDP-1 , where D is a Diagonal matrix and P is an
invertible matrix of eigenvectors.
• Process:
➢ Find Eigenvalues λ
➢ Find Corresponding Eigenvectors X
➢ Construct P from the eigenvectors
➢ D contains the eigenvalues on the diagonal
3. Inner product squares:
• Inner product: 〈u, v〉
• Norm: ‖v‖ = √〈u, v〉
• Orthogonality: u ⊥ v if 〈u, v〉 = 0
〈v,u〉
• Orthogonal Projection: proju v = 〈v,u〉 u
4. Gram-Schmidt process:
Given the Basis {v1 , v2 , . . . , vn }:
• u1 = v1
• u2 = v2 − proju1 v2
• u3 = v3 − proju1 v3 − proju2 v3
Continue similarly for 𝑢𝑛
5. Orthogonal matrices:
• Orthogonal Matrix: Q such that QT Q = QQT = I
• Properties:
BY SHERWIN NAIDOO
1. Eigenvalues and Eigenvectors
• Eigenvalue equation: Ax = λx
• Characteristic Polynomial: 𝑑𝑒𝑡(A − 𝜆I) = 0
• Eigenvalues (λ): Solutions to the characteristic polynomial.
• Eigenvectors (x): refers to non-zero vectors satisfying, Ax = 𝜆x.
2. Diagonalization:
• Diagonalizable matrix: A = PDP-1 , where D is a Diagonal matrix and P is an
invertible matrix of eigenvectors.
• Process:
➢ Find Eigenvalues λ
➢ Find Corresponding Eigenvectors X
➢ Construct P from the eigenvectors
➢ D contains the eigenvalues on the diagonal
3. Inner product squares:
• Inner product: 〈u, v〉
• Norm: ‖v‖ = √〈u, v〉
• Orthogonality: u ⊥ v if 〈u, v〉 = 0
〈v,u〉
• Orthogonal Projection: proju v = 〈v,u〉 u
4. Gram-Schmidt process:
Given the Basis {v1 , v2 , . . . , vn }:
• u1 = v1
• u2 = v2 − proju1 v2
• u3 = v3 − proju1 v3 − proju2 v3
Continue similarly for 𝑢𝑛
5. Orthogonal matrices:
• Orthogonal Matrix: Q such that QT Q = QQT = I
• Properties:
