91Ó°ÊÓ

The characteristic equation is a crucial component when calculating the eigenvalues of a matrix. In simple terms, it is derived from the expression \( \det(A - \lambda I) = 0 \), where \( A \) is the matrix for which you're finding eigenvalues, \( \lambda \) represents the eigenvalues, and \( I \) is the identity matrix of the same size as \( A \). This equation helps us determine the values of \( \lambda \) that satisfy the condition.

• Identity Matrix \( I \): In our example, this is a 2x2 matrix.
• Eigenvalues \( \lambda \): The unknowns we aim to discover.

Here, by setting \( A - \lambda I \) as \( \left[\begin{array}{cc} 1-\lambda & i \ i & 1-\lambda \end{array}\right] \), we will compute its determinant to form our equation. By solving \( \det(A - \lambda I) = 0 \), we are able to extract the characteristic polynomial, which in turn helps us discover the eigenvalues.

The eigenspace is a set of all eigenvectors associated with a particular eigenvalue, including the zero vector. It creates a subspace where each vector satisfies \( (A - \lambda I)\mathbf{v} = 0 \), where \( \mathbf{v} \) is an eigenvector corresponding to \( \lambda \).

• For eigenvalue \( \lambda_1 = 1+i \): We find that the vectors in the eigenspace are solutions to the equation \( \left[\begin{array}{cc} -i & i \ i & -i \end{array}\right]\left[\begin{array}{c} x \ y \end{array}\right] = \left[\begin{array}{c} 0 \ 0 \end{array}\right] \).
• Resulting Vector: Solving gives us the relationship \( x = y \), leading to a basis \( \left\{ \left[\begin{array}{c} 1 \ 1 \end{array}\right] \right\} \).

Similarly, for \( \lambda_2 = 1-i \), we follow the same steps, yielding the basis \( \left\{ \left[\begin{array}{c} 1 \ -1 \end{array}\right] \right\} \). Thus, each eigenvalue has its distinct eigenspace.

Complex numbers extend the concept of one-dimensional real numbers to two-dimensional ones, combining a real part and an imaginary part. The general form of a complex number is \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part with \( i \) denoting \( \sqrt{-1} \).

• Imaginary Unit \( i \): Defined by \( i^2 = -1 \).
• Complex Conjugate: For a complex number \( a+bi \), the conjugate is \( a-bi \), often used in division or simplification.

In our specific exercise, the matrix \( A \) involves the imaginary unit \( i \). The eigenvalues were calculated to be \( 1+i \) and \( 1-i \), showcasing how complex numbers help in understanding rotations and transformations through matrices.

The quadratic formula is instrumental in finding the roots of any quadratic polynomial. It is expressed as \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where the polynomial is structured as \( ax^2 + bx + c = 0 \).

• Discriminant: The part under the square root, \( b^2 - 4ac \), helps determine the nature of the roots. If negative, roots are complex numbers.
• Application: Replace \( a, b, \) and \( c \) with coefficients from the characteristic polynomial for calculations.

In our example, solving \( \lambda^2 - 2\lambda + 2 = 0 \), we input \( a = 1 \), \( b = -2 \), and \( c = 2 \) into the formula, resulting in \( \lambda = 1 \pm i \). This demonstrates the power and utility of the quadratic formula in solving for eigenvalues efficiently.