Matrices
Table of Contents
1. Trace of a Matrix
1.1. Properties
- \[ \tr(AB) = \tr(BA) \neq \tr (A) \tr(B) \]
- \[ \tr(A A^{T}) = \left( \sum_{i=1}^n a_{ii}^2 \right) \geq 0 \]
- \[ \tr(A\pm B) = \tr (A) \pm \tr (B) \]
- \[ \tr(ABC) = \tr(BCA) = \tr(CAB) \]
Special case: If \( A,B,C \) are symmetric matrices, then
\[ \tr(ABC) = \tr(CBA) \] - Trace of product of symmetric matrix (A) and skew-symmetric matrix (B) is zero.
\[ \tr(AB) = 0 \]
2. Eigenvalues and Eigenvectors
For a vector \( A \):
\[ AX = \lambda X \]
\( X \) is known as the eigenvector and the scalar \( \lambda \) is known as the eigen value.
To solve this equation, we can write it as
\[ (A-\lambda I) X = 0 \]
Since, \( X \) is non-singular,
\[ \implies \det (A- \lambda I) = 0 \]
2.1. Properties
- Sum of eigenvalues = Trace of matrix
\[ \det (A-\lambda I) = 0 \]
\[ \sum_i \lambda_i = \tr(A) \] - Product of eigenvalues = Determinant of matrix
\[ \prod_i \lambda_i = \det(A) \] - For a specific eigenvalue, there are infinite eigenvectors, each are scalar multiples of each other.