Do Two Matrices Always Have the Same Determinant If They Have the Same Eigenvalues and Eigenvectors?

It is a common misconception that if two matrices share the same eigenvalues and eigenvectors, they must have the same determinant. This article aims to clarify this misconception by delving into the properties of eigenvalues, eigenvectors, and determinants of matrices. Additionally, it will provide counterexamples to illustrate why the determinant of matrices sharing the same eigenvalues and eigenvectors can vary.

Eigenvalues and Determinants

The determinant of a matrix is closely related to its eigenvalues. Specifically, the determinant of a square matrix is equal to the product of its eigenvalues. For a matrix ( A ) with eigenvalues ( lambda_1, lambda_2, ldots, lambda_n ), the determinant can be expressed as:

[ text{det}(A) lambda_1 cdot lambda_2 cdot ldots cdot lambda_n ]

While this relationship tells us that if two matrices have the same eigenvalues, their determinants are products of the same set of eigenvalues, it doesn’t guarantee that the determinants are identical. This is due to the ordering and multiplicity of the eigenvalues, as well as additional factors that can influence the determinant.

Eigenvectors and Matrix Similarity

The eigenvectors of a matrix are vectors that are only scaled by the matrix. However, having the same eigenvectors does not necessarily mean that the matrices are similar. Similar matrices are matrices that represent the same linear transformation in different bases. Two matrices ( A ) and ( B ) are similar if there exists an invertible matrix ( P ) such that:

[ B P^{-1}AP ]

For similar matrices, the determinants, eigenvalues, and eigenvectors are all the same. However, if two matrices do not share the same similarity transformation, they may have the same eigenvalues and eigenvectors but differ in their structure or scaling factors, leading to different determinants.

Counterexample and Further Considerations

To better illustrate this concept, let’s consider a specific example:

Example: Matrices with Same Eigenvalues and Eigenvectors

Consider the following two matrices:

[ A begin{pmatrix} 1 0 0 2 end{pmatrix} ]

[ B begin{pmatrix} 1 1 0 2 end{pmatrix} ]

Both matrices ( A ) and ( B ) have the eigenvalues ( 1 ) and ( 2 ). However, their determinants are calculated as follows:

[ text{det}(A) 1 cdot 2 2 ]

[ text{det}(B) 1 cdot 2 - 0 cdot 1 2 ]

In this case, both matrices have the same determinant. However, consider another matrix:

[ C begin{pmatrix} 1 0 0 -2 end{pmatrix} ]

Matrix ( C ) also has the eigenvalues ( 1 ) and ( 2 ), but the determinant is:

[ text{det}(C) 1 cdot (-2) -2 )

These examples demonstrate that matrices can have the same eigenvalues and eigenvectors but different determinants if they are not similar or if their eigenvalues differ in sign or multiplicity.

Complex Matrix Example

Even in complex matrix spaces, the determinant can vary if the matrices do not share the same properties. For instance, consider the complex numbers ( i ) and ( 5i ) as matrices acting on a two-dimensional real vector space. Neither matrix has any real eigenvalues or eigenvectors, but their determinants differ by a factor of 25:

[ text{det}(i) -1 ]

[ text{det}(5i) -25 ]

Over an algebraically closed field, while the determinants are related, it’s not generally true that two matrices with the same eigenvalues have the same determinant unless they share additional properties such as similarity.

Conclusion

In conclusion, while matrices sharing the same eigenvalues and eigenvectors can have the same determinant, this is not always the case. Additional factors, such as the similarity of matrices and the multiplicity or sign of eigenvalues, can lead to different determinants. Understanding these concepts is crucial for advanced linear algebra and matrix theory, and it helps in avoiding common misconceptions regarding matrices and their properties.