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

step 1 The topic of this question is eigenvalues and eigenvectors. The question asks us to find a basis

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

Key Concepts

Eigenspace and Basis for Eigenspaces

An eigenspace is the set of all eigenvectors corresponding to a particular eigenvalue, together with the zero vector, forming a subspace of the vector space. Finding a basis for an eigenspace involves selecting a set of linearly independent eigenvectors that span the entire eigenspace. This basis simplifies further analysis and is essential for tasks such as diagonalizing the matrix.

Eigenvectors

Eigenvectors are the nonzero vectors that, when transformed by the matrix A, only change by a scalar factor (the eigenvalue) instead of changing direction. They represent the invariant directions of the transformation, and finding them along with eigenvalues provides insight into the structure and properties of the matrix.

Characteristic Equation

The characteristic equation is a polynomial equation derived by setting the determinant of (A - ?I) equal to zero, where A is the given matrix, ? represents the eigenvalues, and I is the identity matrix of the same dimensions as A. Solving this equation yields the eigenvalues of the matrix, which are crucial in understanding the behavior of linear transformations.

Eigenvalues

Eigenvalues are the scalars ? that satisfy the equation A*v = ?*v for some nonzero vector v. They indicate the factor by which the direction of an eigenvector is scaled during the transformation represented by matrix A. Determining eigenvalues is fundamental to many applications in linear algebra, including stability analysis and diagonalization.

Transcript

00:01 The topic of this question is eigenvalues and eigenvectors.

00:06 The question asks us to find a basis for each eigenspace of the matrix, and thus do all the steps leading up to that, including finding, solving the characteristic equation, and finding the eigenvalues.

00:24 So, what is the characteristic equation? this equation is c of lambda equals zero, where c of lambda is the characteristic polynomial, defined by the determinant of m minus lambda i, where m is what i am calling our matrix, and i as the identity matrix.

00:52 So i want to write out m minus lambda i, and then take its determinant.

01:03 So the determinant is a co -factor expansion along any row or column of the matrix, so i'm going to choose this column, since it has two zeros, and that allows me to, that gives me that i only have to compute the full factor of one entry to be minus lambda.

01:29 Now, since this entry is one of these, five entries of this matrix, whose sum of row and column numbers is even.

01:41 The multiplier of negative one to the power of something will be negative one to the power of, an even number and so multiplier is just one and i'm multiplying this onto the sub matrix that i get or the determinant rather of the sub matrix i get by deleting the row and column of this entry and that is this determinant all the all these entries would have a multiplier of negative one since their sum of row and column numbers is odd.

02:34 Now evaluating this 2x2 determinant, which is ad minus bc, we can evaluate our complete determinant.

02:46 Sorry, this is not equal here.

02:56 Since 8 plus 1 is 9, our complete factorization is lambda minus 3 cubed, since this is lambda minus 3 squared.

03:15 We only have one eigenvalue, one solution to this characteristic equation.

03:25 So we want to find a basis for the eigenspace corresponding to this eigenvalue.

03:34 So now we know that this matrix has one eigenvalue 3...

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