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Eigenvalue calculations

Eigenvalue of a linear transformation

Let \(V\) be a vector space over \(F\) and let \(T:V\to V\) be linear. We say \(\lambda\in F\) is an eigenvalue of \(T\) if

\[ T(v)=\lambda v, \]

for some \(v\in V\backslash\{0\}\). Such \(v\) is called an eigenvector corresponding to the eigenvalue \(\lambda\).

Example （我不知道我亂出的）

Let \(V=\mathsf{P}(\mathbb{R})\), the vector space containing all real polynomials, and let \(T:V\to V\) be defined by

\[ T(f(x))=\dfrac{d}{dx}f(x)+\int_{0}^{1}f(x)dx. \]
  1. Is \(T\) linear?
  2. If \(T\) is linear, find all of the eigenvalues of \(T\).
  3. Is \(T\) injective?
Solution
  1. Yes, \(T\) is linear.
\[ \begin{align*} T(f(x)+c\cdot g(x))&=\dfrac{d}{dx}(f(x)+c\cdot g(x))+\int_{0}^{1}(f(x)+c\cdot g(x))\\ &=\dfrac{d}{dx}f(x)+c\dfrac{d}{dx}g(x)+\int_{0}^{1}f(x)dx+c\int_{0}^{1}g(x)dx\\ &=\left(\dfrac{d}{dx}f(x)+\int_{0}^{1}f(x)dx\right)+c\left(\dfrac{d}{dx}g(x)+\int_{0}^{1}g(x)dx\right)\\ &=T(f(x))+cT(g(x)) \end{align*} \]

Since \(T(1)=0+1=1\), \(1\) is an eigenvalue of \(T\). For \(n\ge 1\), since

\[ T(x^n)=x^{n-1}+\frac{1}{n+1}\neq\lambda x^n, \]

for any \(\lambda\in\mathbb{R}\), the only eigenvalue of \(T\) is \(1\), and \(1\) is an eigenvector corresponding to it.

\(T\) is not injective, since

\[ T(2x-3)=2+\int_{0}^{1}(2x-3)dx=2+(-2)=0=T(0). \]
觀察法

注意

只適用在有限維度！！！

\(A\in\text{M}_{n\times n}(F)\)

  1. 看\(A\)行列式是不是\(0\)。(How?)
  2. 猜一個整數\(k\)，看\(A\)對角線減去\(k\)行列式是不是\(0\)。
  3. 特徵值的和等於trace，特徵值的乘積等於對角線元素乘積，列方程，用上面的\(k\)化成\(2\)次式。
  4. 牛頓有理根檢驗法
公式解
  • 若\(A\in\text{M}_{3\times 3}(F)\)，則
\[ \det(\lambda I-A)=\lambda^3-\text{tr}(A)\lambda^2+S_2\lambda-\det(A) \]
  • 若\(A\in\text{M}_{2\times 2}(F)\)，則
\[ \det(\lambda I-A)=\lambda^2-\text{tr}(A)\lambda+\det(A) \]
驗算法

設\(A\in\text{M}_{n\times n}(F)\)

  1. \(\text{nullity}(A-\lambda_i I)\)一定至少是\(1\)，也就是說\(\text{rank}(A-\lambda_i I)<n\)
  2. 檢驗\(\text{tr}(A)=\sum\lambda_i\)
  3. 檢驗\(\prod_{i=1}^{n}a_{ii}=\prod_{i=1}^n\lambda_i\)
  4. 檢驗\(Av=\lambda_iv\)，其中\(v\)是對應到\(\lambda_i\)的特徵向量