Understanding Eigenvalues
Eigenvalues are crucial in understanding matrices and their applications across various fields such as physics, engineering, and computer science. Let's explore some examples:
Example 1
Consider the matrix A:
A = \[\begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}\]
The characteristic equation to find the eigenvalues is:
\(\lambda^2 - 7\lambda + 10 = 0\)
Solving this, we find the eigenvalues \( \lambda_1 = 5 \) and \( \lambda_2 = 2 \).
Example 2
For the matrix B:
B = \[\begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix}\]
Characteristic equation:
\(\lambda^2 + 3\lambda + 2 = 0\)
The eigenvalues are \( \lambda_1 = -1 \) and \( \lambda_2 = -2 \).
Feel free to experiment with more matrices to find their eigenvalues and explore deeper into linear algebra.