Consider the eigenvalue decomposition of a symmetric matrix a. prove that two eigenvectors vị and v; associated with two distinct eigenvalues li and l; of a are mutually orthogonal; that is, v. vj = 0

Lets consider the symmetric matrix , and the two eigenvectors and such as:

with

.

The dot product between and can be obtained with:

Using the first eigenvector equation we can find:

Lets transpose it

as is an scalar

Now, as A is symmetric:

so

Lets take the dot product again:

but this is :

But, the parenthesis is equal to

so

Now, subtracting the dot product

As the eigenvalues are distinct,   can't be 1, so

this implies

so the eigenvectors are orthogonal.

answer: options

step-by-step explanation: