Problem 29 - Advanced Linear Algebra

Question

Suppose that \(F\) is algebraically closed. Prove that if \(A\) has eigenvalues \(\lambda_{1}, \ldots, \lambda_{n}\) and \(B\) has eigenvalues \(\mu_{1}, \ldots, \mu_{m}\), both lists including multiplicity, then \(A \otimes B\) has eigenvalues \(\left\\{\lambda_{i} \mu_{j} \mid i \leq n, j \leq m\right\\}\), again counting multiplicity.

Step-by-Step Solution

Answer

The eigenvalues of the Kronecker product (tensor product) \(A \otimes B\) are the products of the eigenvalues of \(A\) and \(B\), as demonstrated by computing the product \((A \otimes B)(v_i \otimes w_j)\) and showing that \((A \otimes B)(v_i \otimes w_j) = (\lambda_i \mu_j) (v_i \otimes w_j)\), where \(\lambda_i\) and \(\mu_j\) are the eigenvalues of \(A\) and \(B\), respectively, and \(v_i \otimes w_j\) are the tensor product of their eigenvectors.

Step 1: Define the Kronecker product

The Kronecker product (tensor product) of two matrices \(A \in \mathbb{F}^{n\times n}\) and \(B \in \mathbb{F}^{m \times m}\) is a block matrix defined as follows: \[ (A \otimes B)_{(i - 1)m + x, (j - 1)m + y} = a_{ij} b_{xy} \] where \(a_{ij}\) are the entries of matrix A, and \(b_{xy}\) are the entries of matrix B.

Step 2: Eigenvectors and eigenvalues of A and B

Given that \(A\) has eigenvalues \(\lambda_1, \ldots, \lambda_n\) and \(B\) has eigenvalues \(\mu_1, \ldots, \mu_m\), there exist eigenvectors \(v_1, \ldots, v_n\) of A and \(w_1, \ldots, w_m\) of B such that: \[ A v_i = \lambda_i v_i , \quad B w_j = \mu_j w_j, \] for \(i \leq n, j \leq m\).

Step 3: Compute the eigenvectors of \(A \otimes B\)

Consider the tensor product of the eigenvectors of \(A\) and \(B\), defined by \[ v_i \otimes w_j = \begin{pmatrix} v_{i1} w_j \\ \vdots \\ v_{in} w_j \end{pmatrix}, \] where \(v_{ik}\) is the k-th entry of eigenvector \(v_i\). Now, compute the product \((A \otimes B)(v_i \otimes w_j)\) as follows: \[ \begin{aligned} (A \otimes B)(v_i \otimes w_j) &= (A \otimes B)\begin{pmatrix} v_{i1} w_j \\ \vdots \\ v_{in} w_j \end{pmatrix} \\ &= \begin{pmatrix} a_{11}B(v_{i1}w_j) & \cdots & a_{1n}B(v_{in}w_j) \\ \vdots & \ddots & \vdots \\ a_{n1}B(v_{i1}w_j) & \cdots & a_{nn}B(v_{in}w_j) \end{pmatrix} \\ &= \begin{pmatrix} a_{11}(\lambda_iv_{i1})w_j & \cdots & a_{1n}(\lambda_iv_{in})w_j \\ \vdots & \ddots & \vdots \\ a_{n1}(\lambda_iv_{i1})w_j & \cdots & a_{nn}(\lambda_iv_{in})w_j \end{pmatrix} \\ &= \begin{pmatrix} \lambda_i v_{i1} a_{11}w_j & \cdots & \lambda_i v_{in} a_{1n}w_j \\ \vdots & \ddots & \vdots \\ \lambda_i v_{i1} a_{n1}w_j & \cdots & \lambda_i v_{in} a_{nn}w_j \end{pmatrix} \\ &= \lambda_i \begin{pmatrix} v_{i1} a_{11}w_j & \cdots & v_{in} a_{1n}w_j \\ \vdots & \ddots & \vdots \\ v_{i1} a_{n1}w_j & \cdots & v_{in} a_{nn}w_j \end{pmatrix} \\ &= \lambda_i (v_i \otimes a_1w_j) \\ &= \lambda_i (\lambda_1 v_i \otimes \mu_j w_j) \\ &= (\lambda_i \mu_j) (v_i \otimes w_j) \\ \end{aligned} \] We have shown that \((A \otimes B)(v_i \otimes w_j) = (\lambda_i \mu_j) (v_i \otimes w_j)\), which means that the eigenvalues of \(A \otimes B\) are indeed the products of the eigenvalues of A and B, as stated in the exercise, with their eigenvectors being the tensor product of the eigenvectors of \(A\) and \(B\). This completes the proof.

Key Concepts

Kronecker Product

The Kronecker product, also known as the tensor product, is a mathematical operation on two matrices that results in a block matrix. This operation is not just regular matrix multiplication but a way to construct a new larger matrix which embodies the characteristics of the original two matrices.

It is particularly useful in theoretical studies because it preserves properties of the original matrices, such as eigenvalues, in a structured manner. The operation is defined as follows: Given two matrices,  A and B, the Kronecker product  A ⨂ B is a matrix where each element  a_{ij}  of matrix A is multiplied by the entire matrix B to create block entries. This results in a matrix of size  (n×m, n×m)  given that  A  is  n×n  and  B  is  m×m.

Mathematically, the entry in block position (i, j) of  A ⨂ B is given by the outer product of the corresponding entries of A and B, meaning, (A ⨂ B)_{(i - 1)m + x, (j - 1)m + y} = a_{ij} b_{xy}, where  a_{ij}  are the entries of A, and  b_{xy}  are the entries of B.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, describing properties of matrices that are invariant under linear transformations. An eigenvalue of a matrix is a scalar that, when we multiply it by a corresponding non-zero eigenvector and subtract this product from the matrix acting on the eigenvector, yields zero.

More formally, if A is a matrix, λ is an eigenvalue, and v is a corresponding eigenvector, then Av = λv. These concepts are pivotal in many areas of physics and engineering, such as stability analysis, vibrations, and quantum mechanics, because they provide insight into the behavior of systems modeled by matrices.

In the context of the Kronecker product, the eigenvalues of the product matrix A⨂B can be directly calculated from the eigenvalues of A and B, which leads to a significant simplification in many applications. For instance, if A and B are matrices with eigenvalues λ_1, ..., λ_n and μ_1, ..., μ_m, respectively, then the eigenvalues of A⨂B are given by the pairwise products of the eigenvalues from A and B, that is, {λ_iμ_j | i ≤ n, j ≤ m}.

Algebraically Closed Fields

An algebraically closed field is a fundamental concept within abstract algebra, particularly within the field of polynomial equations. A field F is considered algebraically closed if every non-constant polynomial in F[x], the set of polynomials with coefficients in F, has at least one root in the field F. This means that all polynomial equations can be solved within the field without the need to extend the field to find solutions.

The complex numbers ℂ are a classic example of an algebraically closed field. In such fields, it is guaranteed that every matrix has a full set of eigenvalues within the field, which is an essential property when discussing the eigenvalues of tensors, as in our exercise.

The importance of the concept in the context of our original problem is that it ensures that for any matrix over an algebraically closed field, all eigenvalues exist within the field, thus simplifying the study of their behavior and the proof of certain theorems, such as the one detailing the eigenvalues of the Kronecker product.