Eigenvector#

線性運算的特徵向量 / eigenvalue of a linear operation

矩陣的特徵向量 / eigenvalue of a matrix

Eigenvalue#

The characteristic polynomial is defined as below.

• $f(t) = a_n t^n + a_{n-1}t^{n-1} + \dots + a_1t + a_0 I_n$
• $f(t) = \det(A-tI_{n})$

Eigenvalues must satisfy the following equation of $t$ (characteristic equation):

Multiplicity#

Multiplicity of $\lambda$: the maximum $k$ for the term $(t-\lambda)^{k}$ in the characteristic polynomial.

Theorem. The dimension of the eigenspace must not be greater than its multiplicity.

Diagonalizable#

A matrix $A$ is called diagonalizable if there exists

• a diagonal matrix $D$
• an invertible $P$ such that

Corollary.

Propositions about Diagonalization#

• The set of column vectors of $P$ is a basis of $\mathbb{R}^{n}$.
• The set of column vectors of $P$ consists of eigenvectors of $A$.
• Entries of $D$ consists of eigenvalues corresponding to column vectors of $P$.

Theorem. For any two eigenvectors, if their corresponding eigenvalues are distinct, they are linearly independent.

Diagonalizability#

1. The number of eigenvalues, taking multiplicity into consideration, is equal to the dimension.
2. Dimension of eigenspace = multiplicity

Theorem.

• Linear operation $T$ is diagonalizable if and only if
• there exists a basis $\mathcal{B}$ such that $[T]_{\mathcal{B}}$ is diagonal.

Note: $[T]_{\mathcal{B}}=B^{-1}AB$.