91Ó°ÊÓ

A nonsingular matrix, also commonly known as an invertible or non-degenerate matrix, is a matrix that has an inverse. In other words, if we have a matrix \( B \), it is nonsingular if there exists another matrix \( B^{-1} \) such that:

• \( B \times B^{-1} = I \)

where \( I \) is the identity matrix. The identity matrix is a special matrix with ones on the diagonal and zeros elsewhere.
One key property of nonsingular matrices is that they do not have zero as an eigenvalue. This characteristic ensures that their determinant is non-zero:

• \( \det(B) eq 0 \)

A matrix being nonsingular implies that we can solve the system of linear equations \( Bx = y \) uniquely, because multiplication by \( B^{-1} \) will isolate \( x \):

• \( x = B^{-1}y \)

Thus, in the context of the given exercise, knowing \( B \) is nonsingular means transformations involving \( B \) are reversible.

A quadratic form involves an expression where the coefficients correspond to a particular matrix \( A \). For a vector \( x \), the quadratic form derived from the matrix \( A \) is:

• \( x^T A x \)

where \( x^T \) is the transpose of \( x \), turning it into a column vector if it was initially a row vector, and \( A \) is a symmetric matrix.
This expression evaluates to a scalar, and it's typically used to determine various properties about the matrix \( A \), such as positive definiteness. If for every non-zero vector \( x \), we have a positive value, then the matrix is positive definite:

• \( x^T A x > 0 \)

Quadratic forms appear in various fields such as optimization, physics, and statistics, especially to represent energy functions or to find minimum and maximum values under given conditions.
In our problem, since \( A \) is positive definite, any quadratic form based on \( A \) will continue maintaining positive values, providing a foundation for the transformations applied with other matrices like \( B \).

Matrix multiplication is a fundamental operation in linear algebra involving the multiplication of two matrices. To multiply two matrices \( A \) and \( B \), the number of columns in \( A \) must match the number of rows in \( B \).
The resulting matrix will have dimensions determined by the number of rows in \( A \) and the number of columns in \( B \). The entries in the resulting product matrix are calculated using the dot product of rows from the first matrix and columns from the second:

• \([AB]_{ij} = \sum_{k} A_{ik}B_{kj}\)

This operation is crucial in various transformations, such as the discussed transformation \( BA B^T \) in the exercise. Here, \( B^T \) denotes the transpose of \( B \), which flips the matrix over its diagonal, turning rows into columns and vice versa.
Matrix multiplication is not commutative, which means that \( AB eq BA \) in general. This property is crucial to understanding transformations correctly, especially in sequences like \( BA B^T \), where order plays a significant role in the outcome.
Understanding matrix multiplication allows one to comprehend how transformations using matrices like \( B \) affect original matrices such as \( A \), leading to results like \( BA B^T \) being positive definite when starting with conditions like \( A \) being positive definite and \( B \) being nonsingular.