91Ó°ÊÓ

To find the eigenvalues of a matrix, we use something called the characteristic polynomial. This polynomial arises from the characteristic equation, which is defined as:

• \[ \det(A - \lambda I) = 0 \]

In this context, \( A \) is our matrix, \( \lambda \) is a scalar, and \( I \) is the identity matrix. By subtracting \( \lambda I \) from \( A \) and finding the determinant, we end up with a polynomial equation in terms of \( \lambda \). This polynomial helps us in determining the potential eigenvalues of the matrix.
For our given matrix, calculating the determinant gives us a cubic polynomial. Solving this polynomial reveals what the eigenvalues are. Once we have the eigenvalues, we can use them to explore deeper into matrix properties and transformations.

Sometimes, the roots of the characteristic polynomial are not real numbers. In such cases, we encounter complex eigenvalues. Complex eigenvalues occur when the discriminant of the quadratic part of our polynomial is negative.
In our problem, besides the real eigenvalue \( \lambda = 1 \), the complex eigenvalues \( 2 \pm 3i \) are derived from the quadratic:

• \( \lambda^2 - 4\lambda + 13 = 0 \)
• Using the quadratic formula, we find that the solutions are complex numbers: \( 2 \pm 3i \).

These complex numbers represent rotations and scalings in a plane. They are crucial in understanding processes like oscillations in systems modeled by matrices.

Once we have the eigenvalues of a matrix, the next step is to find the eigenspaces. The eigenspace of an eigenvalue \( \lambda \) is the set of all non-zero vectors, \( \mathbf{v} \), that satisfy:

• \[ (A - \lambda I)\mathbf{v} = 0 \]

For each eigenvalue, we find the null space of the matrix \( A - \lambda I \). These vectors are solutions to the equation above.
For our real eigenvalue \( \lambda = 1 \), the eigenspace is computed by simplifying and solving \( (A - I)\mathbf{v} = 0 \).
Similarly, for the complex eigenvalues \( 2 + 3i \) and \( 2 - 3i \), the corresponding eigenspaces are found by solving for the null space in each instance. This often involves working with complex conjugates in matrix form.

The quadratic formula offers a way to find solutions to the quadratic equation \( ax^2 + bx + c = 0 \). It is a powerful tool, especially when dealing with characteristic polynomials that result in quadratics. The formula is:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our exercise, the characteristic polynomial involves a quadratic part, \( \lambda^2 - 4\lambda + 13 \), leading us to use the quadratic formula to find its roots. The calculation reveals that the discriminant, \( b^2 - 4ac = 16 - 52 = -36 \), is negative, indicating complex solutions.
This tells us that the eigenvalues are \( 2 \pm 3i \). The quadratic formula thus not only helps find these values but also provides insight into the nature (real or complex) of the solutions.