Question

If \(A\) and \(B\) are positive definite, show that \(\left[\begin{array}{cc}A & 0 \\\ 0 & B\end{array}\right]\) is positive definite.

Short answer

The matrix is positive definite because it is symmetric and has positive eigenvalues from \(A\) and \(B\).

Step-by-step solution

Understand Definitions

A matrix is positive definite if it's symmetric and all its eigenvalues are positive. Both matrices \(A\) and \(B\) are given as positive definite, so they must fulfill these criteria individually.

Formulate the Matrix

The matrix given is \(M = \left[\begin{array}{cc}A & 0 \ 0 & B\end{array}\right]\). It is a block diagonal matrix where \(A\) and \(B\) form the diagonal blocks.

Check Symmetry

Since \(A\) and \(B\) are symmetric matrices, the matrix \(M\) will also be symmetric. A block diagonal matrix is symmetric if its blocks are symmetric.

Find Eigenvalues

For block diagonal matrices, the eigenvalues are the union of the eigenvalues of \(A\) and \(B\). Since both \(A\) and \(B\) are positive definite, all their eigenvalues are positive.

Conclusion

Since \(M\) is symmetric and all eigenvalues (from \(A\) and \(B\)) are positive, \(M\) is positive definite.

Key concepts

Eigenvalues

Eigenvalues are crucial in understanding the properties of matrices, especially when it comes to determining whether a matrix is positive definite. Simply put, the eigenvalues of a matrix are the special numbers that describe how much a matrix stretches or compresses vectors along its eigenvectors. In the context of defining a positive definite matrix, all eigenvalues need to be positive. This is vital because the positivity of eigenvalues ensures that a matrix, when multiplied by any non-zero vector, yields a strictly positive value for the quadratic form associated with that vector. For our given matrices, each eigenvalue of the blocks contributes to the positivity of the eigenvalues of the entire block diagonal matrix.

• The eigenvalues reflect the intrinsic scale of the transformation described by the matrix.
• For a matrix to be positive definite, all its eigenvalues must be strictly greater than zero.

In summary, if all eigenvalues of a block are positive, and the matrix is symmetric, it helps in confirming the matrix is positive definite.

Symmetric Matrices

A symmetric matrix is a matrix that is equal to its transpose. This means that the elements are mirrored along the main diagonal, creating a beautifully balanced matrix. The symmetry of a matrix is especially important when discussing positive definite matrices because to consider a matrix as positive definite, it must first be symmetric. Symmetry ensures that the eigenvalues of the matrix are real, which is a cornerstone property needed to evaluate its definiteness. For the matrix given in our problem, since both matrices\( A \) and \( B \) are symmetric, their combined block diagonal matrix will inherently be symmetric as well.

• All entries mirror along the diagonal.
• Symmetry is necessary for eigenvalues to be real numbers.

This symmetry of the matrix simplifies the process of evaluating other properties, such as determining its eigenvalues.

Block Diagonal Matrices

Block diagonal matrices are a type of matrix where non-zero elements are confined to square blocks along the diagonal, while all other entries are zero. In simpler terms, it's composed of smaller square matrices (or blocks) aligned along the main diagonal, like the configuration of the matrix \( M \) in our problem. These matrices are significant because they simplify many operations; their eigenvalues are simply the collection of the eigenvalues of the individual blocks. Thus, the positive definiteness or other properties of the entire matrix rely heavily on the properties of these smaller blocks.

• The operations on block diagonal matrices can be simplified by independently considering each block.
• The structure of a block diagonal matrix aids in efficient computation and the simplicity of analyzing its properties.

For instance, if each block is individually positive definite, the whole matrix by nature of their block arrangement becomes positive definite, as seen in our given matrix \( M \).