Show that if A ∈ℂ^p ×q and B ∈ℂ^q ×p, then det(λIp-A B)=λ^p-q det(λIq-B A) if λ≠0 .

Step 1: Consider the matrices $A in mathbb{C}^{p times q}$ and $B in mathbb{C}^{q times p}$. The

Show that if A ∈ℂ^p ×q and B ∈ℂ^q ×p, then det(λIp-A...

Show that if $A \in \mathbb{C}^{p \times q}$ and $B \in \mathbb{C}^{q \times p}$, then
$$
\operatorname{det}\left(\lambda I_p-A B\right)=\lambda^{p-q} \operatorname{det}\left(\lambda I_q-B A\right) \text { if } \lambda \neq 0 .
$$

Key Concepts

Determinant and Matrix Size Relationship

The determinant of a matrix is a scalar value that reflects certain properties like invertibility and eigenvalues. In the context of matrices of different sizes connected through products (such as A in C^(p×q) and B in C^(q×p)), the difference in sizes leads to additional factors in the determinant formula. Specifically, the factor ?^(p-q) accounts for the extra dimensions when p > q, thus modifying the determinant to reflect the influence of zero eigenvalues that arise from the size disparity.

Sylvester's Determinant Theorem

Sylvester's Determinant Theorem is a result that relates the determinants of matrices of the form I + AB and I + BA, where A and B are matrices of appropriate sizes. Although the theorem is traditionally stated for matrices added to the identity matrix, it underpins the general idea that products of matrices can have determinant identities that bridge different dimensions. The theorem provides foundational insight into why the determinants of ?I - AB and ?I - BA are closely related up to a power of ?.

Nonzero Eigenvalue Correspondence

When dealing with products of matrices such as AB and BA, an important concept is that they share the same nonzero eigenvalues. This property arises from the fact that if ? is a nonzero eigenvalue of AB, then ? is also an eigenvalue of BA. This correspondence is pivotal when comparing the characteristic polynomials of these matrices, as it explains the relationship between the determinants after accounting for potential zero eigenvalues due to dimensional differences.

Schur Complement

The Schur complement is a technique used in linear algebra to compute the determinant of a block matrix. When a matrix is partitioned into blocks, the determinant of the matrix can often be expressed in terms of the determinant of one of the blocks and the Schur complement of that block. This method is especially useful when one of the blocks (or its complement) is invertible, as it allows for the reduction of the determinant calculation to lower-dimensional matrices.

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