Let each matrix in Exercises 1-6 act on ℂ^2 . Find the eigenvalues and a basis for each eigenspa

Let each matrix in Exercises 1-6 act on ℂ^2 . Find the...

Key Concepts

Characteristic Polynomial

The characteristic polynomial is derived from the determinant of the matrix (A - ?I) and its roots are the eigenvalues of the matrix. It is a fundamental tool in linear algebra for understanding the eigenstructure of a matrix.

Eigenvalues

Eigenvalues are the scalars for which there exists a nonzero vector (an eigenvector) such that when the matrix acts on that vector, the result is the same as scaling the vector by the eigenvalue. They are typically found by solving the characteristic equation, which involves determining when the determinant of (A - ?I) is zero.

Eigenspaces

An eigenspace is the set of all eigenvectors corresponding to a particular eigenvalue, along with the zero vector. This subspace is found as the null space of (A - ?I), which means it is the collection of all vectors that are mapped to the zero vector when (A - ?I) is applied.

Complex Vector Spaces

When matrices act on complex vector spaces, such as C^2, the eigenvalues and eigenvectors can be complex. The theory of eigenvalues and eigenvectors applies in this context similarly as in the real case, allowing for a complete decomposition even when complex numbers are involved.

Transcript

00:01 In this example, we have a 2x2 matrix a that's provided, and our goal here is to find its eigenvalues and corresponding eigenvectors.

00:08 To start off for the first part, to find the eigenvalues, we take the determinant of a minus lambda i, where we'll be solving for the variable lambda.

00:19 As a determinant, we'll have negative lambda, negative 8 in the first column, then 1 and 4 minus lambda in the second column.

00:27 Then to take this to determine it, we'll first have the main diagonal multiplied, which is negative lambda, times 4 minus lambda, and subtract the product of the off diagonal, which is negative 8, so we'll have a positive 8 so far.

00:44 Then if we simplify this, we obtain lambda squared minus 4 lambda plus 8 equals 0, and this is our characteristic equation.

00:57 The solutions are lambda 1 equals to 2 minus 2 lambda, or excuse me 2i, and lambda 2 is its conjugate.

01:08 2 with a plus 2i.

01:11 So we can find that using the quadratic formula, and the next thing we're going to do is determine the corresponding eigenvectors for each eigenvalue.

01:20 So let's focus on lambda 1 to start out.

01:23 We can solve the system a minus lambda 1 which is 2 minus 2i times x equals the zero vector so let's augment and say our matrix is going to be distributing the negative sign in in the first column we'll have negative 2 plus 2i and a negative 8 below in the second column we have 1 then take 4 and add together the result of taking negative 2 plus 2i and we'll have altogether 2 plus 2i then we're augmenting with the zero vector which i'll place here now let's do two operations as we row reduce i want to go to the second row divide it by negative 8 then make it become the first row we'll have 1 then then negative .25 plus .25 i and a 0.

02:25 Then the second row is going to be the old first row, which is negative 2 plus 2i, 1 and a 0.

02:33 Now we have a pivot that's here and our goal is to eliminate this entry.

02:39 So let's start by copying the first row.

02:42 It is a 1, negative .25 plus .25i and a 0 .05 and a 0.

02:50 And what we'll be doing is multiplying row 1 by 2 minus 2i and adding the results to the second row.

02:57 We'll obtain 0, and if we take negative 0 .25 plus 0 .25i and multiply it by 2 minus 2i, we obtain exactly negative 1...

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