91Ó°ÊÓ

In linear algebra, an eigenvalue is a special scalar associated with a linear transformation matrix and its eigenvectors. To better understand eigenvalues, think of them as numbers that characterize how a matrix transformation stretches or shrinks vectors in specific directions. When a matrix \( A \) acts on its eigenvector \( \mathbf{v} \), it simply scales \( \mathbf{v} \) by a factor called the eigenvalue, denoted as \( \lambda \).

For a given matrix \( A \) and its eigenvector \( \mathbf{v} \), the relationship can be expressed as:

• \( A\mathbf{v} = \lambda\mathbf{v} \)

This equation highlights that the effect of \( A \) on \( \mathbf{v} \) does not change its direction—only its magnitude is altered.

• If \( \lambda > 1 \), the eigenvector is stretched.
• If \( \lambda < 1 \), the eigenvector is compressed.
• If \( \lambda = 1 \), the vector maintains its length.
• If \( \lambda = 0 \), the vector collapses to zero.

Eigenvalues play a crucial role in diverse applications, from solving differential equations to analyzing stability in dynamical systems. In this exercise, we explore how subtracting a scalar from the identity matrix affects the eigenvalues of a transformed matrix.

Scalar multiplication is a basic operation in vector space where a vector is multiplied by a scalar (a real number). When talking about matrices, each entry of a matrix or vector is multiplied by the scalar, resulting in stretching or compressing the matrix or vector. In mathematical terms, if you have a vector \( \mathbf{v} \) and a scalar \( c \), scalar multiplication is expressed as:

• \( c\mathbf{v} = (cv_1, cv_2, \ldots, cv_n) \)

where \( v_1, v_2, \ldots, v_n \) are the components of the vector.

In the context of the exercise, the expression \( -cI\mathbf{v} \) showcases scalar multiplication where all identities of the matrix \( I \), called the identity matrix, are multiplied by \( -c \). This leads to scaling down or up the eigenvectors depending on the sign and magnitude of \( c \). Such maneuver allows us to derive the new eigenvalue as shown:

• (\(A - cI\))\(\mathbf{v} = A\mathbf{v} - cI\mathbf{v} \)
• \(= \lambda\mathbf{v} - c\mathbf{v} \)
• \(= (\lambda - c)\mathbf{v} \)

Thus, scalar multiplication serves as a bridge in transforming the eigenvalue \( \lambda \) to \( \lambda - c \), emphasizing the fundamental rule in manipulating eigenvalue problems.

An identity matrix, often denoted by \( I \), plays a crucial role in linear algebra as it is analogous to the number 1 in multiplication. The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. A 2x2 identity matrix appears as:

• \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \)

It has a unique property that any matrix \( A \) multiplied by the identity matrix results in the same matrix:

• \( AI = IA = A \)

This property is essential for preserving the matrix's attributes when performing mathematical operations.

In the exercise, we utilize the identity matrix to manipulate the original matrix \( A \) by subtracting \( cI \). The role of the identity matrix here is to ensure that each element of \( A \) is adjusted by \( c \), allowing for the examination of the new eigenvalue. Considering \( A - cI \), each eigenvalue of \( A \) is shifted by \(-c\), resulting in the modified eigenvalue \( \lambda - c \). The identity matrix thus provides a straightforward mechanism for shifting eigenvalues uniformly across the matrix, highlighting its versatility in linear transformations.