Question

Let \(A=\left[\begin{array}{cc}B & 0 \\ 0 & C\end{array}\right]\) where \(B\) and \(C\) are square matrices. a. Show that \(c_{A}(x)=c_{B}(x) c_{C}(x)\). b. If \(\mathbf{x}\) and \(\mathbf{y}\) are eigenvectors of \(B\) and \(C,\) respectively, show that \(\left[\begin{array}{l}\mathbf{x} \\\ 0\end{array}\right]\) and \(\left[\begin{array}{l}0 \\\ \mathbf{y}\end{array}\right]\) are eigenvec- tors of \(A,\) and show how every eigenvector of \(A\) arises from such eigenvectors.

Short answer

a. \(c_A(x)=c_B(x) c_C(x)\). b. Eigenvectors of \(A\) are formed from eigenvectors of \(B\) and \(C\).

Step-by-step solution

Understanding the Block Matrix Structure

The matrix \(A\) is defined as \(A=\begin{bmatrix} B & 0 \ 0 & C \end{bmatrix}\), where both \(B\) and \(C\) are square matrices. This structure is crucial because it simplifies the computation of the determinant and characteristic polynomial of \(A\).

Deriving the Characteristic Polynomial of A

The characteristic polynomial \(c_A(x)\) of a matrix \(A\) is computed as \(\det(xI - A)\), where \(I\) is the identity matrix of appropriate size. For a block matrix, this becomes \(\det\begin{bmatrix} xI_B - B & 0 \ 0 & xI_C - C \end{bmatrix}\), which simplifies to \(\det(xI_B - B) \cdot \det(xI_C - C)\). Therefore, \(c_A(x) = c_B(x) c_C(x)\).

Establishing Eigenvectors of A from B and C

Consider \(\mathbf{x}\) as an eigenvector of \(B\) with eigenvalue \(\lambda\). Then, \((B - \lambda I_B)\mathbf{x} = 0\). Form a vector for \(A\) as \(\begin{bmatrix} \mathbf{x} \ 0 \end{bmatrix}\). Thus, \((A - \lambda I_A)\begin{bmatrix} \mathbf{x} \ 0 \end{bmatrix} = \begin{bmatrix} (B - \lambda I_B)\mathbf{x} \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\). Therefore, \(\begin{bmatrix} \mathbf{x} \ 0 \end{bmatrix}\) is an eigenvector of \(A\).

Applying the Same for C

Similarly, if \(\mathbf{y}\) is an eigenvector of \(C\) with eigenvalue \(\mu\), then \((C - \mu I_C)\mathbf{y} = 0\). Construct the vector \(\begin{bmatrix} 0 \ \mathbf{y} \end{bmatrix}\) for \(A\). The computation \((A - \mu I_A)\begin{bmatrix} 0 \ \mathbf{y} \end{bmatrix} = \begin{bmatrix} 0 \ (C - \mu I_C)\mathbf{y} \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\), shows \(\begin{bmatrix} 0 \ \mathbf{y} \end{bmatrix}\) is an eigenvector of \(A\).

Concluding Eigenvector Representation of A

Every eigenvalue of \(A\) will correspond either to \(B\) or \(C\). All eigenvectors of \(A\) can be formed from the eigenvectors of \(B\) or \(C\), taking the forms \(\begin{bmatrix} \mathbf{x} \ 0 \end{bmatrix}\) and \(\begin{bmatrix} 0 \ \mathbf{y} \end{bmatrix}\) respectively.

Key concepts

Characteristic Polynomial

The characteristic polynomial of a matrix is a fundamental concept in linear algebra that provides valuable insights into the matrix's properties. For a given square matrix \( A \), the characteristic polynomial is denoted by \( c_A(x) \) and is computed using the formula \( \det(xI - A) \), where \( I \) is the identity matrix of the same size as \( A \). This polynomial helps in finding the eigenvalues of the matrix, which are the roots of this polynomial.

In the context of block matrices, such as \( A = \begin{bmatrix} B & 0 \ 0 & C \end{bmatrix} \), the characteristic polynomial takes a particularly intriguing form. Here, \( B \) and \( C \) are square matrices themselves. The structure of the block matrix allows for a simplification in the calculation: the characteristic polynomial of \( A \) can be expressed as the product of the characteristic polynomials of \( B \) and \( C \).

More formally:

• Calculate \( c_B(x) \) as \( \det(xI_B - B) \).
• Calculate \( c_C(x) \) as \( \det(xI_C - C) \).
• The characteristic polynomial of \( A \) is \( c_A(x) = c_B(x) \cdot c_C(x) \).

This property underscores the decomposable nature of block matrices, providing an efficient way to understand their eigenvalues and other properties.

Eigenvectors

Eigenvectors are essential in understanding how matrices operate on spaces. For any square matrix \( M \), an eigenvector \( \mathbf{v} \) associated with eigenvalue \( \lambda \) satisfies the equation \( M\mathbf{v} = \lambda\mathbf{v} \). This relationship implies that applying the matrix \( M \) to \( \mathbf{v} \) stretches or compresses it by a factor of \( \lambda \) without altering its direction.

When dealing with block matrices, the eigenvectors of the overall matrix \( A = \begin{bmatrix} B & 0 \ 0 & C \end{bmatrix} \) can be constructed from the eigenvectors of the submatrices \( B \) and \( C \). If \( \mathbf{x} \) is an eigenvector of \( B \) with eigenvalue \( \lambda \):-

• Then the vector \( \begin{bmatrix} \mathbf{x} \ 0 \end{bmatrix} \) becomes an eigenvector of \( A \).
• Similarly, if \( \mathbf{y} \) is an eigenvector of \( C \) with eigenvalue \( \mu \), the vector \( \begin{bmatrix} 0 \ \mathbf{y} \end{bmatrix} \) is an eigenvector of \( A \).

Every eigenvector of \( A \) can be represented in these forms, derived straightforwardly from the eigenvectors of the constituent matrices \( B \) and \( C \). This reflects how block matrices preserve the foundational features of their components, making analysis more efficient.

Determinant Calculation

The determinant of a matrix is a scalar value that offers critical information about the matrix, such as whether it is invertible and the volume scaling it describes in transformation. To compute the determinant of a block matrix like \( A = \begin{bmatrix} B & 0 \ 0 & C \end{bmatrix} \), you can make use of its block diagonal structure, resulting in a simplified calculation.

For block matrices where off-diagonal blocks are zero, the determinant of the whole matrix \( A \) is simply the product of the determinants of the blocks on the diagonal:

• Calculate \( \det(B) \), the determinant of one block.
• Calculate \( \det(C) \), the determinant of the other block.
• The determinant of \( A \) is \( \det(A) = \det(B) \times \det(C) \).

This approach takes advantage of the block structure to avoid unnecessary complications. It offers a straightforward method to ascertain the properties of the larger matrix based on its smaller, more manageable constituents. Understanding determinants in this way is especially useful in applications involving transformation and volume-related properties in higher-dimensional spaces.