Let A R33 Show that A has at least one real eigenvalue Hint

Let A R3×3. Show that A has at least one real eigenvalue. (Hint: The characteristic polynomial is of degree 3. Can you show that any degree three polynomial with coecients in R has a real root?

Solution

In graphical view,
Each 3x3 matrix represents a 3rd degree polynomial charecteristics.
The 3 eigen values of the matrix are the 3 roots of the polynomial
There cannot be a 3rd degree polynomial without real roots.Try plotting a cubic without crossing X - axis.Hence There should be atleast one real eigen value for a 3x3 matrix.

or:

Consider a scalar matrix Z, obtained by multiplying an identity matrix by a scalar; i.e., Z = c*I. Deducting this from a regular matrix A gives a new matrix A - c*I.

Equation 1: A - Z = A - c*I.

If its determinant is zero,

Equation 2: |A - c*I| = 0

and A has been transformed into a singular matrix. The problem of transforming a regular matrix into a singular matrix is referred to as the eigenvalue problem.

However, deducting c*I from A is equivalent to substracting a scalar c from the main diagonal of A. For the determinant of the new matrix to vanish the trace of A must be equal to the sum of specific values of c. For which values of c?