Matrices
- [Australian National University]
- Overview
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices can have any number of rows and columns.
Matrices are used to represent mathematical objects or their properties. They are useful for describing systems of linear or differential equations, and representing a linear application.
- A matrix is a rectangular array of numbers representing data, used for efficient storage and manipulation of information in rows and columns.
- Square matrices have the same number of rows and columns.
- The order of a matrix describes the number of rows and columns in the matrix.
- A symmetric matrix is a square matrix with reflection symmetry across the main diagonal.
- The identity matrix is an matrix that has zeros everywhere except along the main diagonal which has ones.
- Transpositions occur when the rows of a matrix are turned into the columns of a new matrix, denoted by the superscript T.
- Triangular matrices are square matrices with either all entries below the diagonal (lower triangular) or all entries above the diagonal (upper triangular) being zero.
- Diagonal matrices have all entries except those along the diagonal as zero.
- Matrices and Linear Algebra
Matrices are the foundation of linear algebra, which is the “language” of most machine learning and analytics algorithms. Data represented in vectors and matrices enables efficient, faster computation, access to linear algebra techniques, and better communication.
Matrices can be used to represent samples with multiple attributes in a compact form. Matrices can also be used to represent linear equations in a compact and simple fashion. Linear algebra provides tools to understand and manipulate matrices to derive useful knowledge from data
Matrices have wide applications in engineering, physics, economics, and statistics, as well as in various branches of mathematics. They are used to create graphs, statistics, calculate and conduct scientific studies and research in a variety of subjects. Matrices are also used to represent real-world statistics such as population, infant mortality rate, and so on.
Matrices are usually named by a capital letter: A, B, C, and so on. Each entry in a matrix is referred to as a ij, such that represents the row and represents the column. Matrices are often referred to by their dimensions: m × n indicating m rows and n columns.
Different operations can be performed on matrices such as addition, scalar multiplication, multiplication, transposition, etc..
Please refer to the following for more details:
- Wikipedia: Matrix.
- Matrices in Data Science
Matrices are a fundamental concept in data science that underpin many mathematical and computational operations. They are used to:
- Store data: Matrices are an efficient way to store tabular data, especially when all the data is numeric.
- Represent information: Matrices provide a structured way to represent information, making it easier to process and extract meaningful insights.
- Model systems: Matrices are used to model systems where the order in which the system generated its outcomes is irrelevant.
- Feature extraction: Matrices are used in feature extraction techniques such as Principal Component Analysis (PCA) and Singular Value Decomposition (SVD). These methods transform high-dimensional data into a lower-dimensional space using matrix operations.
- Represent statistics: Matrices are used to represent real-world statistics such as population, infant mortality rate, and so on.
- Portray asset portfolios: The financial domain utilizes matrices to portray asset portfolios, assess the risk and return of investments, and execute operations like asset allocation and optimization.
- Matrices in AI
Matrices are a fundamental concept in artificial intelligence (AI) and machine learning. They are used for: data representation, modeling, training algorithms, manipulating data, modeling parameters, performing mathematical operations.
In deep learning, matrices are used to represent the input data and the weights in a neural network.
Matrices are used in many sub-fields of machine learning, including: image processing and synthesising, natural language processing, prediction.
Matrices and matrix multiplication can also:
- Provide a compact representation of data
- Facilitate efficient computations
- Enable the application of linear algebraic methods to uncover patterns and relationships in the data
The rank of a matrix is the maximum number of its linearly independent column vectors (or row vectors). The rank of a matrix cannot exceed the number of its rows (or columns).
- Matrices in Deep Learning
Matrices are used in deep learning for a number of reasons, including:
- Data representation: Matrices can represent data in a compact way, which can facilitate efficient computations.
- Linear algebraic methods: Matrices and matrix multiplication can be used to uncover patterns and relationships in data.
- Neural networks: Neural networks, which are a core part of deep learning, rely on matrix operations to update weights, propagate information, and make predictions.
- Image processing: Matrices can represent images, and operations like convolution and pooling can be used to downsample images and extract features.
- Model parameters: Matrices can be used to represent and manipulate data, and perform various mathematical operations.
- Structured representation: Matrices provide a structured and organized way to represent information, making it easier to process and extract meaningful insights.
In summary, the use of matrices and matrix multiplication in machine learning and recommender systems provides a compact representation of data, facilitates efficient computations, and enables the application of powerful linear algebraic methods to uncover patterns and relationships in the data.
- Matrices in Neural Networks
In a neural network, matrices are used to represent the input data and the weights assigned to each piece of data. The input data is organized into a matrix, which is then multiplied by the weight matrix to calculate the weighted sum of the input data.
In machine learning, datasets and images are often represented as matrices. Each row in a dataset corresponds to a sample, and each column represents a feature or attribute of the data. This tabular representation allows for efficient storage and processing of large datasets.
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