Determine all eigenvalues and corresponding eigenvectors of the given matrix. [ 2 -1

Determine all eigenvalues and corresponding eigenvectors of...

Key Concepts

Eigenvalues

Eigenvalues are scalars associated with a square matrix such that there exists a nonzero vector (called an eigenvector) which, when the matrix is applied to it, merely scales the vector by that scalar. These scalars are found by solving the characteristic equation derived from setting det(A - ?I) = 0.

Eigenvectors

Eigenvectors are the nonzero vectors that satisfy the equation Av = ?v, where A is the given matrix and ? is an eigenvalue. They indicate the directions in which the linear transformation defined by the matrix acts as a simple scaling operation.

Characteristic Polynomial

The characteristic polynomial is formed by computing the determinant of the matrix (A - ?I), where I is the identity matrix and ? represents a scalar parameter. The roots of this polynomial give the eigenvalues, which play a central role in understanding the matrix's behavior.

Eigenspace

The eigenspace corresponding to a particular eigenvalue is the set of all eigenvectors associated with that eigenvalue, together with the zero vector. This subspace is the kernel of (A - ?I) and provides insight into the geometric structure of the matrix transformation.

Transcript

00:01 This problem gives us a matrix and asks us to find its eigenvalues and eigenvectors.

00:05 We do this by finding the characteristic polynomial, which is found by taking the determinant of a minus lambda i.

00:14 That'll give us the determinant of the matrix 2 minus lambda, 3, 2, negative 1, 1 minus lambda, negative 1, 30, 3 minus lambda, which will give us the polynomial 2 minus lambda times 1 minus lambda times 3 minus lambda minus lambda minus 9 minus 6 times 1 minus lambda plus 3 times 3 minus lambda that will simplify out to be lambda times 2 minus lambda times lambda minus 4 set equal to 0, giving us our eigenvalues to be equal to 0, 2, and 4.

00:59 Then taking the eigenvalue of 0, we're going to plug this into the a minus lambda i equation in order to find the eigenvector x that solves for the zero vector.

01:13 We're going to do this for all the eigenvalues.

01:15 So by plugging in 0, we get the matrix 232, negative 1, 1, negative 1, 303 times the eigenvector i'm going to call x1.

01:29 Then we can perform gash and elimination to simplify this matrix order to make solving easier.

01:37 By subtracting the first row to the bottom row, we get 2 -913, 310, and then 0, all 0 is on the bottom row, times x1.

01:52 Then we can set this equal to the 0 vector.

01:57 So this gives us the system of equations 2 times a1 minus b1 plus 3c1 equal to 0, and 3a1 minus b1 equal to 0, which one you solve will give us the components for a1, b1, c1, which will be negative 3, 9, and 5, making this our final answer for this first eigenvector.

02:26 We do the same then for a lambda equal to 2...

Please give Ace some feedback