Question

Let \(A=\left[\begin{array}{cc}B & 0 \\ 0 & C\end{array}\right]\) where \(B\) and \(C\) are square matrices. a. If \(B\) and \(C\) are diagonalizable via \(Q\) and \(R\) (that is, \(Q^{-1} B Q\) and \(R^{-1} C R\) are diagonal), show that \(A\) is diagonalizable via \(\left[\begin{array}{cc}Q & 0 \\ 0 & R\end{array}\right]\) b. Use (a) to diagonalize \(A\) if \(B=\left[\begin{array}{ll}5 & 3 \\ 3 & 5\end{array}\right]\) and \(C=\left[\begin{array}{rr}7 & -1 \\ -1 & 7\end{array}\right]\)

Short answer

Matrix \( A \) is diagonalizable by \( P \), and its diagonal form is \( \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 8 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 8 \end{bmatrix} \).

Step-by-step solution

Check if A is block diagonal

Note that matrix \( A \) is block diagonal, composed of matrices \( B \) and \( C \). This allows us to diagonalize \( A \) by separately diagonalizing \( B \) and \( C \).

Diagonalization condition for B and C

Given that \( B \) and \( C \) are diagonalizable, it means there exist invertible matrices \( Q \) and \( R \) such that \( Q^{-1} B Q \) and \( R^{-1} C R \) are diagonal matrices, say \( D_1 \) and \( D_2 \) respectively.

Construct P for diagonalizing A

Construct the matrix \( P \) as \( P = \left[\begin{array}{cc} Q & 0 \ 0 & R \end{array}\right] \). Then for diagonalization, we need to show \( P^{-1} A P \) is diagonal.

Verify diagonalization of A with P

Compute \( P^{-1} A P \):\[P^{-1} = \left[\begin{array}{cc} Q^{-1} & 0 \ 0 & R^{-1} \end{array}\right]\]\[P^{-1} A P = \left[\begin{array}{cc} Q^{-1} & 0 \ 0 & R^{-1} \end{array}\right] \left[\begin{array}{cc} B & 0 \ 0 & C \end{array}\right] \left[\begin{array}{cc} Q & 0 \ 0 & R \end{array}\right] = \left[\begin{array}{cc} Q^{-1} B Q & 0 \ 0 & R^{-1} C R \end{array}\right]\]This results in a diagonal matrix \( \left[\begin{array}{cc} D_1 & 0 \ 0 & D_2 \end{array}\right] \). Thus, \( A \) is diagonalizable via \( P \).

Diagonalize B and C using eigenvectors and eigenvalues

Calculate eigenvalues and eigenvectors of \( B \):Eigenvalues \(\lambda_1 = 2 \), \( \lambda_2 = 8 \); Eigenvectors are \( \begin{bmatrix} -1 \ 1 \end{bmatrix} \) for \( 2 \) and \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \) for \( 8 \). Construct matrix \( Q = \begin{bmatrix} -1 & 1 \ 1 & 1 \end{bmatrix} \).For \( C \), Eigenvalues are \( 6 \), \( 8 \) with eigenvectors \( \begin{bmatrix} -1 \ 1 \end{bmatrix} \) for \( 6 \) and \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \) for \( 8 \). Construct matrix \( R = \begin{bmatrix} -1 & 1 \ 1 & 1 \end{bmatrix} \).

Form the P matrix for A and verify the diagonalization

Combine the matrices \( Q \) and \( R \) into a block diagonal matrix \( P \): \[P = \begin{bmatrix} -1 & 1 & 0 & 0 \ 1 & 1 & 0 & 0 \ 0 & 0 & -1 & 1 \ 0 & 0 & 1 & 1 \end{bmatrix}\]Verify that \( P^{-1} A P \) results in the diagonal matrix \(\begin{bmatrix} 2 & 0 & 0 & 0 \ 0 & 8 & 0 & 0 \ 0 & 0 & 6 & 0 \ 0 & 0 & 0 & 8 \end{bmatrix}\).

Check that diagonalization is correct

Verify by calculation: \( P^{-1} \) can be computed, and multiplying it with \( A \) and \( P \) stepwise should yield \( \begin{bmatrix} 2 & 0 & 0 & 0 \ 0 & 8 & 0 & 0 \ 0 & 0 & 6 & 0 \ 0 & 0 & 0 & 8 \end{bmatrix} \), confirming diagonalization is correct.

Key concepts

Block Diagonal Matrix

A block diagonal matrix, like matrix \( A \) in our exercise, is a special type of square matrix. It consists of smaller square matrices, known as blocks, positioned along its diagonal. This pattern makes it easier to work with, especially when it comes to operations like diagonalization. Since the blocks do not interfere with each other along non-diagonal paths, each block can be processed independently. This property significantly simplifies computations.

• In our example, matrix \( A \) has two blocks: \( B \) and \( C \). Both are square matrices, and together they form a block diagonal structure in \( A \).
• For matrix \( A \), we take advantage of its block diagonal form by independently diagonalizing \( B \) and \( C \).

Block diagonal matrices are commonly used in various applications because they allow operations to be broken down into smaller, more manageable tasks.

Eigenvectors and Eigenvalues

Eigenvectors and eigenvalues are fundamental in linear algebra, especially for understanding the properties of matrices. They provide insights into transformations represented by the matrix, like stretching, rotating, and scaling.

• Eigenvalues: These are scalars \( \lambda \) such that when a matrix \( M \) is multiplied by a non-zero vector \( \mathbf{v} \) (the eigenvector), the product is scaled by \( \lambda \). Mathematically, \( M \mathbf{v} = \lambda \mathbf{v} \).
• Eigenvectors: Eigenvectors are the vectors that correspond to these eigenvalues. They remain in the same direction after the transformation by the matrix \( M \).

In our exercise, we first find the eigenvalues for matrices \( B \) and \( C \). For \( B \), these are \( 2 \) and \( 8 \) with corresponding eigenvectors \( \begin{bmatrix} -1 \ 1 \end{bmatrix} \) and \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \). Similarly, for \( C \), the eigenvalues are \( 6 \) and \( 8 \). The eigenvectors reflect a fundamental axis of transformation for each respective block.

Diagonal Matrix

A diagonal matrix is one in which all elements outside the main diagonal are zero. Such matrices are particularly simple to work with due to their structure.

• The diagonal elements directly represent the eigenvalues of the matrix when the matrix is diagonalizable.
• Diagonal matrices simplify many matrix operations, such as multiplication and finding powers of matrices.

In the context of our problem, once matrices \( B \) and \( C \) are diagonalized, we obtain diagonal matrices \( D_1 \) and \( D_2 \). Matrices \( D_1 \) and \( D_2 \) are interesting because they capture the scaling effect the original matrices \( B \) and \( C \) apply to their eigenvectors. Furthermore, constructing a block diagonal version of \( D_1 \) and \( D_2 \) gives us a complete diagonal form of \( A \).

Invertible Matrices

Invertible matrices are those that possess a matrix inverse. For a matrix to be invertible, it must be square (having the same number of rows and columns) and its determinant should be non-zero. The existence of an inverse matrix is crucial for performing various matrix operations.

• For diagonalization, having an invertible matrix is essential. The transformation matrix used to diagonalize another matrix must itself be invertible.
• In our exercise, matrices \( Q \) and \( R \) are used to diagonalize matrices \( B \) and \( C \), respectively. Both \( Q \) and \( R \) are invertible, ensuring that the diagonalization is mathematically valid.

When a matrix is invertible, it implies a strong relationship with its eigenvalues and vectors, ensuring transformations can be reversed. This feature is vital when constructing matrix \( P \) using blocks of \( Q \) and \( R \), facilitating the diagonalization of the larger block diagonal matrix \( A \).