V. NumPy Linear Algebra (线性代数)#

The numpy.linalg module provides standard linear algebra (线性代数) operations on 2-D arrays treated as matrices (矩阵). These are essential for machine learning, physics simulations, and engineering calculations.

1. Matrix Multiplication (矩阵乘法)#

Core idea: Use @ or np.dot() for matrix multiplication — NOT * (which is element-wise).

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import numpy as np
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A = np.array([[1, 2], [3, 4]])
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B = np.array([[5, 6], [7, 8]])
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A @ B          # Matrix multiply (矩阵乘法)
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np.dot(A, B)   # Same result: [[19 22] [43 50]]
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A * B          # Element-wise multiply (逐元素乘法) — different!

Note: Always use @ or np.dot() for matrix multiplication. Using * gives element-wise results, which is a common mistake.

2. `linalg.inv()` — Inverse Matrix (逆矩阵)#

Core idea: Find matrix $A^{-1}$ such that $A \cdot A^{-1} = I$ (identity matrix).

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A = np.array([[1, 2], [3, 4]])
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A_inv = np.linalg.inv(A)
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# [[-2.   1. ]
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#  [ 1.5 -0.5]]
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# Verify: A @ A_inv ≈ Identity matrix (单位矩阵)
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np.round(A @ A_inv)   # [[1. 0.] [0. 1.]]

Note: Only square, non-singular matrices (非奇异矩阵) are invertible. A singular matrix (奇异矩阵) will raise LinAlgError.

3. `linalg.det()` — Determinant (行列式)#

Core idea: A scalar value describing matrix properties. If det = 0, the matrix is singular (not invertible).

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A = np.array([[1, 2], [3, 4]])
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np.linalg.det(A)   # -2.0
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B = np.array([[1, 2], [2, 4]])  # rows are proportional
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np.linalg.det(B)   # 0.0  → singular matrix!

4. `linalg.eig()` — Eigenvalues & Eigenvectors (特征值与特征向量)#

Core idea: Find $\lambda$ and $v$ such that $A \cdot v = \lambda \cdot v$ . Core of PCA (主成分分析) and many ML algorithms.

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A = np.array([[4, 1], [2, 3]])
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eigenvalues, eigenvectors = np.linalg.eig(A)
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# eigenvalues:  [5. 2.]
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# eigenvectors: columns are the eigenvectors
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print(eigenvalues)    # [5. 2.]
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print(eigenvectors)   # [[0.707 -0.447]
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                      #  [0.707  0.894]]

5. `linalg.svd()` — Singular Value Decomposition (奇异值分解)#

Core idea: Decompose any matrix $M = U \Sigma V^T$ . Used in data compression, image processing, and recommendation systems.

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A = np.array([[1, 2], [3, 4], [5, 6]])
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U, S, Vt = np.linalg.svd(A)
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# U: left singular vectors (shape: 3×3)
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# S: singular values (shape: 2,) — diagonal of Σ
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# Vt: right singular vectors transposed (shape: 2×2)
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print(S)   # [9.525 0.514]

6. `linalg.solve()` — Solve Linear Equations (线性方程组)#

Core idea: Solve $Ax = b$ for $x$ . More numerically stable than computing $A^{-1} \cdot b$ .

$Ax = b \quad \Rightarrow \quad x = \text{linalg.solve}(A, b)$

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# Solve: 2x + y = 5
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#        x + 3y = 10
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A = np.array([[2, 1], [1, 3]])
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b = np.array([5, 10])
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x = np.linalg.solve(A, b)
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# [1. 3.] → x=1, y=3
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# Verify
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np.allclose(A @ x, b)   # True

Note: Always prefer linalg.solve() over inv(A) @ b for numerical stability and performance.

7. Quick Comparison Table#

Function (函数)	Mathematical Operation	Use Case
`@` / `dot(A,B)`	$AB$	Matrix multiply
`linalg.inv(A)`	$A^{-1}$	Invert matrix
`linalg.det(A)`	$\det(A)$	Check singularity
`linalg.eig(A)`	$Av = \lambda v$	PCA, stability
`linalg.svd(A)`	$U\Sigma V^T$	Compression, rank
`linalg.solve(A,b)`	$Ax = b$	Linear systems

💡 One-line Takeaway
Use @ for matrix multiplication, linalg.solve() instead of inv() for equation solving, and check det() before inverting.

V. NumPy Linear Algebra (线性代数)#

1. Matrix Multiplication (矩阵乘法)#

2. linalg.inv() — Inverse Matrix (逆矩阵)#

3. linalg.det() — Determinant (行列式)#

4. linalg.eig() — Eigenvalues & Eigenvectors (特征值与特征向量)#

5. linalg.svd() — Singular Value Decomposition (奇异值分解)#

6. linalg.solve() — Solve Linear Equations (线性方程组)#