Guozhen AIGlobal AI field notes and model intelligence

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English editions of Guozhen AI articles. The text is localized for global readers while the original diagrams, screenshots, and code examples remain aligned with the Chinese source.

Assume we have historical stock price data

State space models describe how a system evolves over time using matrices. They unify historical states, external inputs, and observed outputs within a single linear fram...

Activation function

At its core, neural network computation still consists largely of matrix multiplications. Understanding tensor shapes, weights, and gradients transforms deep learning fro...

Generate sample data

Machine learning training is commonly expressed in matrix form: a batch of samples is processed simultaneously to compute predictions, followed by parameter updates based...

Load image

The core idea behind SVD applications is to preserve dominant structural components while discarding weak, noisy ones. Image compression, recommendation systems, and PCA...

Define matrix A

You can compute singular values manually by starting from the eigenvalues of $A^T A$. In practice, engineers delegate this task to numerical libraries—but understanding t...

Generate a simple random matrix (imagine it as a grayscale image)

SVD decomposes any matrix into three components: direction, magnitude (strength), and direction again. It is more general than eigenvalue decomposition and better suited...

Extract TF-IDF vectors for the first and second documents

Inner product spaces bring geometric notions—such as distance, angle, and projection—into algorithmic design. Recommendation systems, search engines, and regression model...

Example vectors

An orthogonal basis functions like a set of mutually independent coordinate rulers. When representing vectors using such a basis, projections and reconstructions become t...

Define vectors

The inner product compresses the relationship between two vectors into a single scalar. It simultaneously connects vector length, angle between vectors, and similarity—ma...

Define matrix A

Eigen decomposition breaks down a complex matrix into two intuitive components: direction (encoded in eigenvectors) and scaling (encoded in eigenvalues). When applicable,...

16. Eigenvectors: Definition and Intuition

An eigenvector is not a fixed length arrow—it is a direction . As long as the direction remains unchanged, scalar multiples of the same vector still represent the same ei...

Define the matrix

Eigenvalues quantify how much a matrix scales vectors along certain special directions. They serve as a crucial entry point for understanding dimensionality reduction, sy...

Define coefficient matrix A

A homogeneous system always has the trivial (zero) solution. For a nonhomogeneous system, we must first determine whether a solution exists. If a solution exists, the gen...

13. Gaussian Elimination for Systems of Linear Equations

Gaussian elimination fundamentally simplifies a system of linear equations using equivalent transformations —operations that preserve the solution set while progressively...

Define coefficient matrix and constant vector

A linear equation describes a constraint—on a line, a plane, or in higher dimensional space. When multiple such constraints are combined, they form a system of equations.

Define a 3×3 matrix

When computing determinants, low order matrices can be expanded directly; for higher order matrices, it’s more efficient to use row operations to convert the matrix into...

10 Properties of Determinants

Determinant properties exist to enable faster, more robust computation—and also help determine whether a matrix is singular (degenerate) or whether a linear transformatio...

Determinants: Definition and Intuition

A determinant can be understood as the scaling factor by which a linear transformation changes the volume of space. When the determinant equals zero, space is flattened—i...

Matrix Transposition and Inversion

Transposition is commonly used to adjust orientation and compute inner products; the inverse matrix represents the transformation that “undoes” the original one. In pract...

Matrix Multiplication and Its Properties

Matrix multiplication is not element wise multiplication—it computes dot products between rows and columns. It models feature weighting, coordinate transformations, and m...

Linear Algebra for AI: Matrix Addition and Scalar Multiplication

Matrix addition and scalar multiplication may appear simple at first glance—but they form the foundational starting point for understanding batch feature updates, weight...

Vector and Matrix Operations

Behind every operation rule lies meaning: addition represents composition, scalar multiplication represents scaling, the dot product measures similarity, and matrix multi...

Create a 2-row, 3-column matrix

There are two most practical ways to understand a matrix:

Simulated pixel values (0–255) for an image

A vector can represent either a geometric arrow or a row of features. When reading AI code, it is far more common to convert samples into feature vectors.

Generate synthetic data

Many computations in AI can be expressed in terms of linear algebra: inputs are vectors, parameters are matrices, and training amounts to finding better directions in hig...

1. Introduction to Linear Algebra: Core Concepts

Don’t rush to memorize formulas when learning linear algebra. Instead, think of vectors as data , matrices as transformations , and systems of equations as constraints —t...