Eigenvalues and Eigenvectors

Table of contents

• 1. Properties
• 2. Note that
• 3. Similar Matrix
• 4. Diagonalization
  • (1) Definition
  • (2) iff condition
  • (3) Sketch
  • (4) Uniqueness

---

1. Properties

1. $ \sum \lambda = tr(A) $
2. $ \prod \lambda = det(A) $
3. Symmetric $S$ has real eigenvalues, and can choose orthogonal eigenvectors.
4. $\lambda_1 \ne \lambda_2 \implies \mathbf{x}_1 \cdot \mathbf{x}_2 = 0$ : different values, independent vectors
5. $\lambda_1 = \lambda_2 \xrightarrow{\text{?}}$ independent vectors (might or might not)
6. $\mathbf{x}_i$ : orthogonal $\iff A^TA=AA^T$ (where $A$ : real)

2. Note that

1. (×) $\lambda(A+B) = \lambda(A) + \lambda(B)$
2. (×) $\lambda(AB) = \lambda(A) \times \lambda(B)$

3. Similar Matrix

: different matrix with same eigenvalues

\[\forall B \text{ : invertible,} \quad BAB^{-1} \text{ is similar to } A\]

$ (BAB^{-1})(B\mathbf{x}) = BA\mathbf{x} = B(\lambda\mathbf{x}) = \lambda(B\mathbf{x}) $ : $\lambda$ is also an eigenvalue of $BAB^{-1}$

4. Diagonalization

(1) Definition

\[\exists P, D \quad \text{s.t.} \quad P^{-1}AP = D \quad (P \text{ : invertible, } D \text{ : diagonal})\]

(2) iff condition

: has a full $n$ independent eigenvectors

(3) Sketch

\[AX = X\Lambda \rightarrow A = X\Lambda X^{-1}\] \[A \begin{bmatrix} & & \\ \mathbf{x}_{1} & \cdots & \mathbf{x}_{n} \\ & & \\ \end{bmatrix} = \begin{bmatrix} & & \\ A\mathbf{x}_{1} & \cdots & A\mathbf{x}_{n} \\ & & \\ \end{bmatrix} = \begin{bmatrix} & & \\ \lambda_1\mathbf{x}_{1} & \cdots & \lambda_n\mathbf{x}_{n} \\ & & \\ \end{bmatrix} = \begin{bmatrix} & & \\ \mathbf{x}_{1} & \cdots & \mathbf{x}_{n} \\ & & \\ \end{bmatrix} \begin{bmatrix} \lambda_1 & & \\ & \ddots & \\ & & \lambda_n\\ \end{bmatrix}\]

(4) Uniqueness

Diagonalization is unique upto
① permutation (on diagonal $D$)
② scalar multiplication (on eigenmatrix $X$)

---