91Ó°ÊÓ

The characteristic polynomial of a matrix is an important concept in linear algebra. It helps us find eigenvalues, which are crucial for understanding the behavior of matrices. The characteristic polynomial is defined as \( p(\lambda) = \det(A - \lambda I) \). Here, \( A \) is our original matrix and \( I \) is the identity matrix of the same size. When calculating, you subtract \( \lambda \) from each diagonal element of \( A \) and compute the determinant of this new matrix.

• The characteristic polynomial turns a matrix problem into a scalar polynomial equation.
• To find eigenvalues, set this polynomial equal to zero and solve for \( \lambda \).

In the provided exercise, the matrix \( A \) is transformed into \( A - \lambda I \), and then the determinant is calculated. For the given matrix, the characteristic polynomial is \( (1-\lambda)^2(\lambda^2 - 2\lambda + 1) \). This polynomial provides insights into the matrix's eigenvalues.

An eigenspace is fundamentally a vector space associated with a given eigenvalue. Once you've found an eigenvalue \( \lambda \), the eigenspace consists of all the vectors \( \mathbf{v} \) that satisfy \( (A - \lambda I)\mathbf{v} = \mathbf{0} \). In simpler terms, these are the vectors which, when transformed by the matrix \( A \), get stretched by the factor \( \lambda \), but do not change direction.

• To find the eigenspace, solve the system \( (A - \lambda I)\mathbf{v} = 0 \).
• This usually involves row-reducing the matrix \( (A - \lambda I) \).

For \( \lambda = 1 \) in our problem, solving gives basis vectors \( \begin{bmatrix}1 \, 0 \, 0 \, 0\end{bmatrix} \) and \( \begin{bmatrix}0 \, 1 \, 0 \, 0\end{bmatrix} \). These vectors form the basis of the eigenspace related to the eigenvalue \( 1 \). For \( \lambda = 0 \), the basis vector is \( \begin{bmatrix}0 \, 0 \, 0 \, 1\end{bmatrix} \). Each eigenspace gives us valuable geometric insights about the transformation described by matrix \( A \).

Algebraic multiplicity pertains to how many times an eigenvalue appears as a root of the characteristic polynomial. It is a non-negative integer that conveys information about the solution structure of the associated polynomial equation. Larger algebraic multiplicity indicates more repeated roots.

• Determine algebraic multiplicity by counting the occurrence of an eigenvalue in the polynomial.
• It tells you the total number of times an eigenvalue is repeated mathematically.

In the exercise, \( \lambda = 1 \) has an algebraic multiplicity of 3 because it appears as a root thrice in the factor \((1-\lambda)^2(\lambda^2 - 2\lambda + 1)\). For \( \lambda = 0 \), it occurs only once, so the algebraic multiplicity is 1. Understanding algebraic multiplicity helps predict the structure of the eigenspace.

Geometric multiplicity refers to the number of linearly independent vectors in the eigenspace corresponding to an eigenvalue. It gives insight into the dimensions of the eigenspace - essentially, how many directions the eigenvalue can scale.

• Find it by counting the number of basis vectors in the eigenspace.
• The geometric multiplicity is always less than or equal to the algebraic multiplicity.

For \( \lambda = 1 \), the geometric multiplicity is 2 because there are two basis vectors: \( \begin{bmatrix}1 \, 0 \, 0 \, 0\end{bmatrix} \) and \( \begin{bmatrix}0 \, 1 \, 0 \, 0\end{bmatrix} \). For \( \lambda = 0 \), it has a geometric multiplicity of 1 with a single basis vector \( \begin{bmatrix}0 \, 0 \, 0 \, 1\end{bmatrix} \). This distinction helps understand how solutions to the eigenvalue equation unfold in multiple dimensions.