Can Non Square Matrix Be Invertible

Can a Non-Square Matrix Be Invertible?

The question of whether a non-square matrix can be invertible is a fundamental concept in linear algebra. The short answer is: no, a non-square matrix cannot be invertible. However, understanding why this is true requires a deeper dive into the properties of invertible matrices and their relationship to linear transformations. This article will explore this concept thoroughly, examining the definitions, theorems, and implications surrounding invertible matrices and their dimensions.

Understanding Invertibility

Before we address the core question, let's establish a clear understanding of what it means for a matrix to be invertible. A square matrix A is invertible (also called nonsingular or non-degenerate) if there exists another matrix, denoted as A⁻¹ (A inverse), such that:

A * A⁻¹ = A⁻¹ * A = I

where I is the identity matrix. The identity matrix is a square matrix with ones along the main diagonal and zeros elsewhere. Multiplication by the identity matrix leaves any other matrix unchanged.

The existence of this inverse matrix has profound implications. It signifies that the linear transformation represented by A is both one-to-one (injective) and onto (surjective). Let's unpack what this means:

One-to-one (injective): Each vector in the domain maps to a unique vector in the codomain. No two distinct vectors in the domain are mapped to the same vector in the codomain.
Onto (surjective): Every vector in the codomain is mapped to by at least one vector in the domain. The transformation covers the entire codomain.

A linear transformation that is both one-to-one and onto is called a bijection. Only bijective linear transformations are represented by invertible matrices.

Why Non-Square Matrices Cannot Be Invertible

The core reason why a non-square matrix cannot be invertible lies in the dimensions of the matrices involved. Consider a non-square matrix A. There are two possibilities:

A is a rectangular matrix with more rows than columns (m x n, where m > n): In this case, the linear transformation represented by A maps vectors from an n-dimensional space to an m-dimensional space. Because m > n, the transformation cannot be onto (surjective). There will be vectors in the m-dimensional space that are not mapped to by any vector in the n-dimensional space. This violates the necessary condition for invertibility.
A is a rectangular matrix with more columns than rows (m x n, where m < n): Here, the linear transformation maps vectors from an n-dimensional space to an m-dimensional space. Because m < n, the transformation cannot be one-to-one (injective). Multiple vectors in the n-dimensional space will map to the same vector in the m-dimensional space. This also violates the necessary condition for invertibility.

In both scenarios, the non-square nature of the matrix prevents the linear transformation from being both injective and surjective. Therefore, it's impossible to find an inverse matrix that satisfies the condition A * A⁻¹ = A⁻¹ * A = I. The matrix multiplication simply isn't defined in the necessary manner.

Alternative Approaches for Non-Square Matrices: Pseudoinverse

While a non-square matrix cannot have a true inverse, there are related concepts that can provide some similar functionality. One such concept is the pseudoinverse (also known as the Moore-Penrose inverse). The pseudoinverse exists for all matrices, including non-square matrices, and provides a best-fit solution in a least-squares sense.

The pseudoinverse of a matrix A, denoted as A⁺, satisfies some but not all of the properties of a true inverse:

A * A⁺ * A = A
A⁺ * A * A⁺ = A⁺
(A * A⁺)ᵀ = A * A⁺ (Hermitian)
(A⁺ * A)ᵀ = A⁺ * A (Hermitian)

The pseudoinverse finds application in various areas, including:

Solving overdetermined or underdetermined systems of linear equations: When you have more equations than unknowns (overdetermined) or more unknowns than equations (underdetermined), the pseudoinverse provides a least-squares solution that minimizes the error.
Signal processing and image reconstruction: The pseudoinverse is used to solve ill-conditioned problems, where small changes in the input lead to large changes in the output.
Machine learning: Pseudoinverses are valuable in various machine-learning algorithms, such as linear regression and principal component analysis.

However, it's crucial to remember that the pseudoinverse is not a true inverse. It doesn't satisfy the condition A * A⁺ = A⁺ * A = I for non-square matrices.

Determinants and Invertibility

The determinant is a scalar value calculated from the elements of a square matrix. A fundamental theorem of linear algebra states that a square matrix is invertible if and only if its determinant is non-zero. This provides another perspective on why non-square matrices can't be invertible. The determinant is only defined for square matrices; therefore, the determinant test for invertibility is inapplicable to non-square matrices.

Implications and Applications

The invertibility of a matrix has significant implications across various fields:

Solving linear equations: Invertible matrices are essential for solving systems of linear equations. If the coefficient matrix is invertible, a unique solution exists.
Linear transformations: Invertible matrices represent bijective linear transformations. This means that the transformation is reversible, allowing us to transform back to the original space.
Computer graphics: Invertible matrices are crucial in computer graphics for performing transformations like rotation, scaling, and translation. The inverse matrix allows for reversing these transformations.
Cryptography: Invertible matrices play a crucial role in various cryptographic algorithms, ensuring secure encryption and decryption.

Understanding the invertibility of matrices is fundamental to a solid grasp of linear algebra and its applications. While non-square matrices lack a true inverse, the pseudoinverse offers a valuable alternative for handling certain problems involving non-square systems. The limitations imposed by matrix dimensions are crucial to understanding the properties and applicability of linear algebraic operations. Remember, the concept of invertibility is intricately tied to the bijective nature of the associated linear transformation, a property that is inherently impossible for non-square matrices to possess.