#L08 · Linear Algebra & Manifolds · Lv.4View in 3D Mind-Map

Eigenvalues & Eigenvectors (Av = λv)

Augustin-Louis Cauchy & David Hilbert · 1829 CE

A\mathbf{v} = \lambda\mathbf{v} \iff \det(A - \lambda I) = 0

\sum \lambda_i = \operatorname{tr}(A),\quad \prod \lambda_i = \det(A)

Finds the invariant axes (eigenvectors v) that do not change direction under matrix transformation A, scaling only by their eigenvalues λ—the crown jewel of linear algebra.

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Entry A₁₁

Off-diag b

Entry A₂₂

\lambda^2 - 5.0\lambda + (0.0) = 0 \implies \lambda_1 = 5.00, \; \lambda_2 = 0.00

제1 고윳값 λ₁ (주성분)

5.0000

제2 고윳값 λ₂

0.0000

대각합 tr(A) = λ₁+λ₂

5.00

Prerequisite Path (16 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 2#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 3#L01Vector Operations, L1/L2 Norms & Normalization

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

STEP 4#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 5#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 6#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 7#G022D/3D Euclidean Distance Formula

d(P

STEP 8#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 9#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 10#L02Dot Product & Cosine Similarity

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

STEP 11#L033D Cross Product & Surface Normal Vector

\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y

STEP 12#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 13#A05Vieta's Formulas (Roots & Coefficients)

\alpha + \beta = -\frac{b}{a}

STEP 14#L04Matrix Multiplication & Linear Transformation

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}

STEP 15#L05Determinant & Matrix Inverse

\det(A) = ad - bc

STEP 16#L08Eigenvalues & Eigenvectors (Av = λv)

A\mathbf{v} = \lambda\mathbf{v} \iff \det(A - \lambda I) = 0

🚀 Next Unlocks (Derived Formulas) (1)

#L09Singular Value Decomposition (SVD) & PCA

Geometric & Intuitive Insight

지구본을 회전시킬 때 북극과 남극을 잇는 자전축 위의 점들은 회전 후에도 방향이 바뀌지 않고 제자리에 머무릅니다. 이처럼 선형변환의 본질적인 '뼈대 축'을 찾아내는 것이 고윳값 분해입니다.

Step-by-Step Derivation

Step 1.1단계: Av = λv 를 이항하여 (A - λI)v = 0 으로 정리합니다.
Step 2.2단계: 영벡터가 아닌 해 v ≠ 0 가 존재하려면 행렬 (A - λI)가 역행렬을 갖지 않아야 하므로 특성방정식 det(A - λI) = 0 이 성립해야 합니다.
Step 3.3단계: 2×2 행렬의 경우 λ² - tr(A)λ + det(A) = 0 을 근의 공식으로 풀어 고윳값 λ₁, λ₂ 를 구합니다.

AI & Coding Real-World Use

Google PageRank algorithm, quantum mechanics Hamiltonian energy states (Hψ=Eψ), and structural resonance

#Google PageRank#Quantum Mechanics Hψ=Eψ#Spectral Graph Theory

const tr = a + d; const det = a*d - b*c; const l1 = (tr + Math.sqrt(tr*tr - 4*det))/2;

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