91Ó°ÊÓ

Scalar multiplication of matrices involves multiplying every element of a matrix by a scalar value. This operation is straightforward and essential in matrix algebra. If you have a matrix \( A \) and a scalar \( c \), the resulting matrix is \( cA \), where each entry \( a_{ij} \) of \( A \) is multiplied by \( c \). This means:

• If \( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \) and \( c = 3 \), then \( cA = \begin{pmatrix} 3a_{11} & 3a_{12} \ 3a_{21} & 3a_{22} \end{pmatrix} \).

This operation is distributive and makes it easy to adjust the scale of the actions facilitated by a matrix. When applied to positive definite matrices, like \( A \), and multiplied by a positive scalar \( c \), the resulting matrix \( cA \) remains positive definite. This is because multiplying the transformation represented by \( A \) by a positive scalar does not change its fundamental characteristic of being positive definite, as it only uniformly scales the transformation's effect.

Eigenvalues play a significant role in determining if a matrix is positive definite. A matrix is positive definite if all its eigenvalues (\( \lambda_i \)) are positive. When we talk about matrix \( A \), if \( A \) is positive definite, then any vector \( \mathbf{x} \) will satisfy \( \mathbf{x}^T A \mathbf{x} > 0 \). This implies all eigenvalues of \( A \) are greater than zero.
For instance, if you square a positive definite matrix \( A \) to form \( A^2 \), the eigenvalues of \( A \) which are originally positive will just be squared as well,

• e.g., \( \lambda_i^2 > 0 \).

Thus, \( A^2 \) will also have positive eigenvalues and it will remain symmetric, hence \( A^2 \) is positive definite. Furthermore, if you take the sum \( A + B \) of two positive definite matrices \( A \) and \( B \), the transformation properties of both matrices add up to result in \((A + B)\), ensuring that the matrix remains positive definite because positive eigenvalues from \( A \) and \( B \) combine, keeping all eigenvalues of the sum positive.

Matrix inversion is the process of finding another matrix \( A^{-1} \) such that when it is multiplied by the original matrix \( A \), it results in the identity matrix. This means \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix.
To invert a matrix \( A \), it must be square and must have full rank, meaning all its eigenvalues \( \lambda_i \) are non-zero. For positive definite matrices, all eigenvalues are not only non-zero but positive, guaranteeing invertibility.

• If \( A \) is positive definite, \( A^{-1} \) will also be positive definite.
• The eigenvalues of \( A^{-1} \) become \( 1/\lambda_i \), maintaining positivity.

This implies \( A^{-1} \) will still produce positive outcomes when interacting with any non-zero vector \( \mathbf{x} \), thus conserving its properties of being positive definite.

Symmetric matrices play an essential role in matrix algebra because of their nice mathematical properties. A matrix \( A \) is symmetric if \( A = A^T \), which means the element in the \( i \)-th row and \( j \)-th column is equal to the element in the \( j \)-th row and \( i \)-th column, specifically \( a_{ij} = a_{ji} \).
Symmetric matrices are especially important because:

• They have real eigenvalues.
• Their eigenvectors are orthogonal.
• Positive definite symmetric matrices maintain positive definiteness when manipulated correctly, such as through addition or scalar multiplication with positive scalars.

In our exercise, matrices \( A \) and \( B \) are symmetric, and this fact ensures that when they undergo operations like addition, scalar multiplication, or even inversion when positive definite, they keep properties that make mathematical results easier and more consistent to work with. A key insight is that symmetric matrices are naturally predisposed to such operations while retaining positive definiteness.