行列（Further Pure 2）

The hidden directions in a matrix

• Every matrix transforms space. Some directions are special: the matrix only stretches 拉伸 them, without rotating. These are the eigenvectors 特征向量, and the stretch factors are the eigenvalues 特征值.
• Eigenvalues unlock the structure of a matrix — they're the key to diagonalisation 对角化, stability analysis, and Google's PageRank algorithm.

Matrix equations 矩阵方程

• Three linear equations in three unknowns → a matrix equation $A\mathbf{x} = \mathbf{b}$. A non-singular $A$ gives one unique solution: $\mathbf{x} = A^{-1}\mathbf{b}$.

An eigenvector keeps its direction under the matrix; a general vector is turned

Eigenvalues and eigenvectors

• The characteristic equation 特征方程 $\det(A - \lambda I) = 0$ gives the eigenvalues $\lambda$.
• For each $\lambda$, the eigenvector $\mathbf{v}$ satisfies $A\mathbf{v} = \lambda\mathbf{v}$.
• Multiplying an eigenvector by $A$ only stretches it (by factor $\lambda$), without changing its direction.

Diagonalisation

• You can write $A = QDQ^{-1}$, where $D$ is a diagonal matrix 对角矩阵 of eigenvalues and the columns of $Q$ are the eigenvectors.
• This makes computing $A^n$ easy: $A^n = QD^nQ^{-1}$, and $D^n$ is just the diagonal entries raised to the $n$th power.

Worked example

• $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$. Characteristic equation: $(2-\lambda)^2 - 1 = 0 \Rightarrow \lambda^2 - 4\lambda + 3 = 0 \Rightarrow \lambda = 1$ or $3$.
• For $\lambda = 1$: eigenvector $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$. For $\lambda = 3$: eigenvector $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$.
• A matrix with zero determinant has a singularity (it is singular and has no inverse).

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