Eigenvalues and Determinants: Exploring the Relationship Between Square and Non-Square Matrices

Understanding the fundamental concepts of linear algebra, such as eigenvalues and determinants, is crucial in various fields including engineering, physics, and computer science. This article aims to clarify why det(A - λI) 0 implies that λ is an eigenvalue of matrix A, and why this particular relationship does not apply to non-square matrices. We will also address the specific case of a transformation from a vector space to a different vector space, and the implications of non-square matrices having full rank.

Understanding Eigenvalues and Determinants

In linear algebra, eigenvalues and eigenvectors are central concepts that describe the behavior of linear transformations. If λ is an eigenvalue of a square matrix A, then by definition, there exists a non-zero vector x such that AX λX. Rearranging this equation gives us (A - λI)X 0. Since a non-zero vector x exists, the matrix (A - λI) must be singular, which means it has a determinant of zero. Hence, det(A - λI) 0.

Conversely, if det(A - λI) 0, then the matrix (A - λI) is singular and has a non-trivial null space, implying the existence of a non-zero x such that (A - λI)X 0. This leads to AX λX, confirming that λ is indeed an eigenvalue of A.

Non-Square Matrices and Determinants

Non-square matrices, such as those with unequal dimensions, do not have determinants in the same sense as square matrices. The determinant is defined only for square matrices, which must have the same number of rows and columns to form a square arrangement of numbers. For non-square matrices, the concept of rank becomes more relevant.

For a non-square matrix A of dimensions m x n, the matrix can be considered full rank if and only if det(ATA) ≠ 0, where T denotes the transpose. This condition ensures that the matrix has a non-zero singular value in the singular value decomposition (SVD). If all singular values are non-zero, the matrix is said to have full rank. Importantly, this does not relate directly to the determinant of the original non-square matrix A but rather to the singular values in its SVD.

It is worth noting that there are some generalizations of the determinant concept for rectangular matrices, such as the generalized singular value or the QR decomposition, but these are typically more complex and not as straightforward as the determinant for square matrices.

Eigenvalues and Non-Square Matrices

The relationship between eigenvalues and non-square matrices is different from that for square matrices. Specifically, if A is a non-square matrix, then the concept of a matrix having eigenvalues becomes irrelevant because non-square matrices do not have eigenvalues in the traditional sense.

In the context of transformations from one vector space to a different vector space, a transformation that maps vectors from a space of dimension m to a space of dimension n (where m ≠ n) cannot have eigenvalues. This is because the transformation equation AX λX does not make sense for non-square matrices, as it would imply that X must be a vector in both vector spaces, which is not possible.

Advanced Concepts and Specifics

For advanced readers, it's important to note that while non-square matrices do not have eigenvalues, they still play a crucial role in linear algebra. Singular Value Decomposition (SVD) is one such important tool that is used extensively in many applications, from machine learning to signal processing. The SVD of a matrix A can be expressed as A UΣVT, where U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of A.

The singular values in Σ can be considered as a generalization of eigenvalues for rectangular matrices. They provide a measure of the scaling factors of the transformation represented by A in different directions. The non-zero singular values indicate that the matrix has full rank, and they play a critical role in understanding the behavior of the transformation.

Conclusion

Summarizing the key points, the relationship between eigenvalues and the determinant, det(A - λI) 0, is inherently tied to square matrices. For non-square matrices, while the concept of full rank and singular values is relevant, the notion of eigenvalues does not apply in the same manner.

Eigenvalues and the determinant provide crucial insights into the behavior of square matrices, but for non-square matrices, the analysis shifts to concepts like rank, singular values, and the SVD. Understanding these differences is essential for students and practitioners in fields that rely on linear algebra, including computer science, data science, and engineering.