Matrix As Product Of Elementary Matrices

Expressing Matrices as a Product of Elementary Matrices

This article explores the fascinating concept of representing any invertible matrix as a product of elementary matrices. Understanding this allows for a deeper insight into matrix operations and linear algebra in general. We'll delve into what elementary matrices are, how they relate to elementary row operations, and the process of decomposing a matrix into its elementary factors. This is a powerful tool with applications in various fields, including solving systems of linear equations and understanding matrix invertibility.

What are Elementary Matrices?

Elementary matrices are square matrices that are obtained by performing a single elementary row operation on an identity matrix. There are three types of elementary row operations, and correspondingly, three types of elementary matrices:

• Type I: Swapping two rows: This involves interchanging two rows of the identity matrix. For example, swapping rows 1 and 2 of a 3x3 identity matrix produces:

[[0, 1, 0],
 [1, 0, 0],
 [0, 0, 1]]

• Type II: Multiplying a row by a non-zero scalar: This involves multiplying a single row of the identity matrix by a non-zero constant. Multiplying the second row of a 3x3 identity matrix by 'k' results in:

[[1, 0, 0],
 [0, k, 0],
 [0, 0, 1]]

• Type III: Adding a multiple of one row to another: This involves adding a multiple of one row to another row in the identity matrix. Adding 'k' times row 1 to row 2 of a 3x3 identity matrix gives:

[[1, 0, 0],
 [k, 1, 0],
 [0, 0, 1]]

The Connection to Elementary Row Operations

The crucial link between elementary matrices and row operations lies in their effect on other matrices. When an elementary matrix multiplies another matrix on the left, it performs the corresponding elementary row operation on that matrix. This is the key to expressing any invertible matrix as a product of elementary matrices.

Expressing an Invertible Matrix as a Product of Elementary Matrices

The process involves applying Gaussian elimination to reduce the invertible matrix to the identity matrix. Each elementary row operation performed during the reduction corresponds to an elementary matrix. By multiplying these elementary matrices together in the reverse order of their application, we obtain the factorization.

Let's illustrate with a simple example:

Consider the matrix A:

A = [[2, 1],
     [1, 1]]

To reduce A to the identity matrix, we can perform the following row operations:

1. Swap rows 1 and 2: This corresponds to elementary matrix E₁.
2. Subtract 2 times row 1 from row 2: This corresponds to elementary matrix E₂.
3. Subtract row 2 from row 1: This corresponds to elementary matrix E₃.
4. Multiply row 2 by -1: This corresponds to elementary matrix E₄.

After these operations, we obtain the identity matrix. Then:

E₄E₃E₂E₁A = I

Therefore, A = E₁⁻¹E₂⁻¹E₃⁻¹E₄⁻¹

The inverses of the elementary matrices are also elementary matrices, and this gives the decomposition of A as a product of elementary matrices.

Significance and Applications

This decomposition is significant for several reasons:

• Understanding Invertibility: A matrix is invertible if and only if it can be expressed as a product of elementary matrices.
• Solving Linear Systems: This decomposition can simplify solving systems of linear equations.
• Computing the Inverse: The product of the inverses of the elementary matrices gives the inverse of the original matrix.
• Theoretical Foundations: It provides a deeper understanding of the structure of invertible matrices within linear algebra.

By understanding how to express a matrix as a product of elementary matrices, we gain a powerful tool for analyzing and manipulating matrices, solidifying our understanding of fundamental linear algebra concepts. This technique provides a more intuitive and structural approach to various matrix computations and is fundamental to many advanced topics in linear algebra and its applications.