Find all eigenvalues and eigenvectors of the given matrix....

Key Concepts

Eigenvalues

Eigenvalues are the scalars that represent the factors by which the eigenvectors are scaled during a linear transformation. They are determined by solving a polynomial equation known as the characteristic equation, where the eigenvalues are found as the roots of the characteristic polynomial derived from the matrix.

Eigenvectors

Eigenvectors are the non-zero vectors that remain in the same direction when a linear transformation is applied, except for being scaled by their corresponding eigenvalues. They are found by solving the homogeneous linear system (A - ?I)x = 0 for each eigenvalue ?.

Characteristic Polynomial

The characteristic polynomial is an essential tool in finding the eigenvalues of a matrix. It is obtained by calculating the determinant of the matrix subtracted by a scalar multiple of the identity matrix, denoted as det(A - ?I). The roots of this polynomial equation provide the eigenvalues of the matrix.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible, and plays a crucial role in forming the characteristic polynomial used to find eigenvalues.

Transcript

00:01 This question covers topic relating to the eigenvalues and eigenvectors of a matrix.

00:08 We are going to find the eigenvalues and eigen flag factor, right? so we are going to find eigen pairs for this matrix.

00:21 So i will let this matrix is a, it called a.

00:25 Then first thing you need to do is find the characteristic polynomial, so i denoted by p lambda.

00:32 So it's going to be the determinant.

00:34 Of a minus lambda i so it's going to be the determinant of 5 minus lambda negative 1 3 1 minus lambda so now you if you know the formula of the determinant you take this multiply by this multiplication and then you minus this right okay so that equivalent to 5 minus lambda 1 minus lambda minus of negative 1 times 3 and then you simplify everything okay that simply gives you lambda square minus 6 lambda and then plus 5 and then plus 3 so it's going to be lambda square minus lambda plus 8 right then you have two solutions right one solution equal to 2 another solution equal to 4 you can use the quadratic formula to five for it then you have lambda 1 equal to 2 and lambda 2 equal to 4 and that is the those are the eigenvectors for this matrix and you are going to find the eigenvalues sorry this is eigenvalues and now you're going to find the eigenvectors for the case lambda 1 equal to 2 so that means you have to find x and y set that a minus 2 i times the vector x1 x2 right equal to 0 right so that equivalent to the matrix here is 3 negative 1 3 and then 1 minus 2 is negative 1 x1 x2 equal to 0 0 right so that equivalent to 3 x1 minus 2 equal to 0 0 0 0 right so that equivalent to 3 x1 minus x 2 equal to 0 0 0 0 0 0 0 0 0 0 0 and 3 x1 minus x2 equal to 0...

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