Find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the matrix.

Find the characteristic equation, the eigenvalues, and bases...

Key Concepts

Basis for Eigenspaces

A basis for an eigenspace consists of a set of linearly independent eigenvectors that span the entire eigenspace. Finding a basis is crucial because it provides a minimal and efficient description of the invariant subspace associated with a given eigenvalue. Such bases are used in various applications, including the diagonalization of matrices and simplifying many linear algebra problems.

Eigenspaces

For each eigenvalue, the eigenspace is the set of all eigenvectors corresponding to that eigenvalue, along with the zero vector. Mathematically, the eigenspace is the null space of the matrix (A - ?I). Determining the eigenspace associated with each eigenvalue involves solving a system of linear equations, which yields a subspace that characterizes the directions invariant under the matrix transformation.

Characteristic Equation

The characteristic equation of a matrix is formed by taking the determinant of (A - ?I), where A is the matrix, ? is a scalar, and I is the identity matrix. Setting this determinant equal to zero produces a polynomial equation in ?, known as the characteristic polynomial. This equation is central in determining the eigenvalues of the matrix, because its roots are exactly those eigenvalues.

Eigenvalues

Eigenvalues are scalar values that satisfy the equation Ax = ?x for some nonzero vector x, known as an eigenvector. They are obtained by solving the characteristic equation and provide insight into the behavior of the matrix, such as whether it stretches or compresses vectors along certain directions. In many applications, eigenvalues are used to analyze systems, stability, and transformations.

Transcript

00:01 We're going to find the characteristic equation, eigenvalues, and bases for the eigenspaces.

00:07 So to find the characteristic equation, we want to solve the determinant of lambda identity minus the matrix.

00:21 Let's call it a.

00:26 So that's the determinant of lambda 0, 0, 0, lambda 0, 0, 0, lambda minus a.

00:38 Okay, so that's the determinant of lambda, negative 1, negative 1, negative 1, lambda, negative 1, negative 1, negative 1, lambda.

00:51 Okay, i'm going to expand along the first row.

00:53 We're going to be lambda times the determinant.

00:57 If i cross out this row and this column, i'm left with this 2 by 2, lambda, negative 1, negative 1, lambda.

01:06 Okay, now we alternate signs.

01:07 So it's going to be minus negative 1, so that's plus 1, times, and then the determinant, if i cross out this row and this column, i'm left with negative 1, negative 1, negative 1 lambda, okay, and then plus a negative 1, if i cross out that row and that column, i'm left with this 2 by 2, negative 1 lambda, negative 1, negative 1, okay, so this is lambda times the determinant of 2 by 2 is the product of the diagonal minus the product of the antideg, so it's going to be lambda squared, minus one.

01:47 Okay we have negative lambda minus one and we have one minus negative lambda will be plus lambda.

02:02 So this is lambda cubed minus lambda minus lambda minus one minus one minus lambda that's lambda cubed minus three lambda minus two.

02:17 Okay so possible integer roots are plus minus one and plus minus two.

02:24 I try one first.

02:26 It's going to be one minus three minus two.

02:28 That's not zero.

02:29 I try negative one.

02:30 It's going to be negative one plus three minus two is zero.

02:34 So negative one is the root.

02:36 I'm going to do synthetic division.

02:38 Negative one.

02:39 The coefficients are one.