91Ó°ÊÓ

Symmetric matrices play a crucial role in linear algebra, particularly in the study of quadratic forms and eigenvalues. A matrix is symmetric if it is equal to its transpose. In mathematical terms, a matrix \(A\) is symmetric if \(A = A^T\). This implies that across the main diagonal, the entries are mirrored. Symmetric matrices are important because they often appear in physical applications where they represent systems with balanced properties.
For example, in the given problem where \(A = LL^T\), proving that \(A\) is symmetric is straightforward using the matrix transposition property. We do this by showing that \(A^T = (LL^T)^T = L^T L = A\), confirming that \(A\) is indeed symmetric.

The transposition property of matrices is essential for understanding symmetric matrices. The transpose of a matrix \(A\), denoted \(A^T\), is obtained by swapping rows with columns. If \(A\) is an \(m \times n\) matrix, \(A^T\) will be an \(n \times m\) matrix. Transposition has several important properties:

• \((A^T)^T = A\)
• \((A + B)^T = A^T + B^T\)
• \((AB)^T = B^T A^T\)

In the context of the given problem, we used the property \((LL^T)^T = (L^T)^T L^T\). Knowing that \((L^T)^T = L\), it simplified to \(L L^T\), showing that the matrix \(A\) remains the same when transposed, proving symmetry.

A matrix \(A\) is said to be positive definite if for any non-zero vector \(x\), the quadratic form \(x^T A x > 0\). Positive definiteness is crucial in optimization problems and ensuring system stability. To demonstrate that a matrix is positive definite, one typically checks that:

• All its eigenvalues are positive
• The given quadratic form is always positive for any non-zero vector \(x\)

In the problem at hand, we prove \(A = LL^T\) is positive definite by substituting \(y = L^T x\), leading to the form \(y^T y\), which is always positive for non-zero \(y\). Thus, \(x^T (LL^T) x > 0\), showing \(A\) is positive definite.

A matrix is nonsingular if it is invertible, meaning there exists another matrix that when multiplied with it, results in the identity matrix. Nonsingular matrices have several key properties:

• The determinant of a nonsingular matrix is non-zero
• They have full rank, meaning their rows and columns are linearly independent
• They exhibit stable and predictable behavior in systems

In the problem's context, the matrix \(L\) is given as real and nonsingular. Since \(L\) is nonsingular, \(L^T\) is also nonsingular, ensuring that the transformation \(L^T x = y\) does not result in a zero vector unless \(x\) is zero. This property is fundamental in proving both the symmetry and positive definiteness of the matrix \(A\).