EIGEN VECTORS OF A MATRIX

Example :

Find the eigenvectors of

To find the eigenvectors, first we have to find the eigen values.

To find the eigen values, solve the following equation.

det|A - λI| = 0

where I is the 2 × 2 identity matrix.

The characteristic equation :

(5 - λ)(2 - λ) - 18 = 0

10 - 5λ - 2λ + λ² - 18 = 0

λ² - 7λ - 8 = 0

(λ - 8)(λ + 1) = 0

λ = 8  or  λ = -1

The two eigenvalues of the given matrix :

λ₁ = 8

λ₂ = -1

An eigenvector of a 2 × 2 matrix is going to be a column vector with 2 components. If we call these two unknowns x and y, then we can write the vector as

For the first eigenvalue, the eigenvector equation is

Av = λ₁v

Specifically, we have the matrix equation,

From the above matrix equation, we have

The linear equations (1) and (2) above represent the same line. So, the system has infinitely many solutions.

Solve (1) or (2) for y.

y = (³⁄₂)x

If we choose x = 2,

y = (³⁄₂)(2)

y = 3

Then the eigenvector corresponding to λ₁ = 8 is

For the second eigenvalue, the eigenvector equation is

Av = λ₂v

Specifically, we have the matrix equation,

From the above matrix equation, we have

The linear equations (1) and (2) above represent the same line. So, the system has infinitely many solutions.

Solve (1) or (2) for y.

y = -3x

If we choose x = 1,

y = -3(1)

y = -3

Then the eigenvector corresponding to λ₁ = -1 is

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