Linear Algebra Theorem and Equations

Eigenvector Equation

Ax = λx⃗

Dependence & Eigenspace

Independence: Let S = {v1, v2, ... , vn}. S is independent if none of the vectors in S can be written as a linear

combination of other vectors in S.

Eigenspace: S is a basis for the eigenspace for an eigenvalue λ if S is independent and every eigenvector x⃗ can be

written as a linear combination of vectors in S.

Eigenvalues and Matrix Entries

Theorem 5.1.2: If A is triangular then the eigenvalues are the diagonal entries of A.

Theorem 5.1.4: (TFAE) A is invertible if and only if λ = 0 is not an eigenvalue of A.

Diagonalization

If A and B are n × n matrices, then they are similar if there is an invertible n × n matrix P, such that

B = P−1AP. If they are indeed similar, then the following are the same for both A and B:

a) Determinant

b) Trace

c) Characteristic Polynomial det(λI − A)

d) Eigenvalues

e) Invertibility

Geometric and Algebraic Multiplicity

Algebraic Multiplicity: the number of times that (λ-λ0) appears as a factor in the characteristic polynomial of A.

Geometric Multiplicity: The number of vectors in a basis for the eigenspace corresponding to λ0. The number of linearly independent eigenvectors associated with an eigenvalue.

Power of a Matrix

If A is diagonalizable and k is a positive integer, then A

k = PDkP−1

(since all middle PP−1 terms cancel out), which is much easier to compute because (Dk)

ii = (Dii) k

Complex Addition

Like vectors, (a + bi) + (c + di) = (a + c) + (b + d)i.

Complex Multiplication

 (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac − bd) + (ad + bc)i.

Complex Division

Z1/Z2 * (barZ2/barZ2)

Polar Form

z = r(cosθ + isinθ)

z = re^iθ

DeMoivre Theorem 

z^n = r^n * e^inθ

Cauchy-Shwartz Theorem

If u⃗⃗ and v⃗ are in R^n then |u⃗⃗ ∙ v⃗| ≤ ‖u⃗⃗‖‖v⃗‖.

Dot Product

u⃗⃗ ∙ v⃗ = ‖u⃗⃗‖‖v⃗‖ cos θ

Triangle Inequality

If u⃗⃗ and v⃗ are in R^n then ‖u⃗⃗ + v⃗‖ ≤ ‖u⃗⃗‖ + ‖v⃗‖.

Properties of the Dot Product

u ⋅ v = v ⋅ u

u ⋅ (v + w) = u ⋅ v + u ⋅ w

k(u ⋅ v) = (ku) ⋅ v

v ⋅ v >= 0 and v ⋅ v = 0 if and only if v = 0

Properties of Cross Product

a) u⃗⃗ × v⃗ = −(v⃗ × u⃗⃗)

b) u⃗⃗ × (v⃗ + w⃗⃗) = u⃗⃗ × v⃗ + u⃗⃗ × w⃗⃗

c) u⃗⃗ × u⃗⃗ = u⃗⃗ × 0⃗⃗ = 0⃗⃗

d) k(u⃗⃗ × v⃗) = ku⃗⃗ × v⃗ = u⃗⃗ × kv

e) u x 0 = 0 x u = 0

f) u x u = 0

Geometric Cross Product Properties

|| u x v ||² = ||u||²||v||² - (u ⋅ v)² (Lagrange’s Identity)

• || u x v || sin theta = area of a parallelogram

|| u x v ||² = ||u||²||v||² - ||u||²||v||² cos² theta (Replacing (u ⋅ v)²)

|| u x v ||² = ||u||²||v||²(1-cos²theta) (Factoring)

|| u x v ||² = ||u||²||v||²sin² theta (Trig Identity)

Lagrange’s Identity 

‖u⃗⃗ × v⃗‖^2 = ‖u⃗⃗‖‖v⃗‖ − (u⃗⃗ ∙ v⃗)^2

Scalar Triple Product

u⃗⃗ ∙ (v⃗ × w⃗⃗) = 

|u1 u2 u3|

|v1 v2 v3|

|w1 w2 w3|

its the volume of the parallelopiped formed by u⃗⃗, v⃗, and w⃗⃗.

Dot and Cross Product Properties

u ⋅ (u x v) = 0 (u x v is orthogonal to u)

v ⋅ (u x v) = 0 (u x v is orthogonal to v)

|| u x v ||² = ||u||² ||v||²  - (u ⋅ v)² (Lagrange’s identity)

u x (v x w) = (u ⋅ w)v - (u ⋅ v)w (Vector triple product)

(u x v) x w = (u ⋅ w)v - (v ⋅ w)u (Vector triple product)