hw problem

Characteristic equation

The equation used to find the eigenvalues of a matrix, given by det(A - λI) = 0.

Eigenvalues symbol

Represented by λ in the characteristic equation.

Steps to find eigenvalues

Subtract λI from the matrix A and calculate the determinant of the resulting matrix, setting it equal to zero.

Meaning of eigenvalues

Each eigenvalue indicates how much a corresponding eigenvector is stretched or compressed during the transformation represented by the matrix.

Finding eigenspace

To find the eigenspace, solve the equation (A - λI)v = 0 for the eigenvector v.

Equation for eigenspace

The equation solved to find the eigenspace is (A - λI)v = 0.

Basis for eigenspace at λ = 0

The basis can be represented as the vector of the form (x, 0, -x). A specific basis example is (1, 0, -1).

Relationship determination in eigenspace

Determined from the system of equations obtained by solving (A - λI)v = 0.

Importance of eigenvalues and eigenspaces

They provide insight into the behavior of linear transformations and are essential in applications like stability analysis.

Significance of eigenvector

Represents a direction where the transformation associated with the matrix acts as a simple scaling operation.

Diagonalizability indicator

A matrix is diagonalizable if it has a complete set of linearly independent eigenvectors.

Construction of matrix P

Constructed using the eigenvectors as its columns.

Role of eigenvectors in matrix P

They form the basis for transforming the matrix into its diagonal form.

Formation of diagonal matrix D

Created by placing the eigenvalues on the diagonal and zeros elsewhere.

Equation A = PDP^{-1}

Represents the relationship between the original matrix A, its eigenvectors P, and its eigenvalues D.

Importance of diagonalization

It simplifies matrix operations and reveals fundamental characteristics of the transformation.