Verify Eigenpairs

Mathematica, 10 bytes

#2.#==#3#&

Takes input like {vector, matrix, scalar} and returns a boolean.

MATL, 7 bytes

*i2GY*=

Inputs in order: l,v,A.

Explanation:

*  % implicitly get l and v, multiply.
i  % get A
2G % get second input, i.e., v again
Y* % perform matrix multiplication
=  % test equality of both multiplications

Surprisingly long answer, if you ask me, mostly because I needed a way to get all the input correctly. I do not think that less than 5 bytes is possible, but it would be cool if someone found a 5 or 6 byte solution.

Basically, this calculates l*v==A*v.

Jelly, 5 bytes

æ.⁵⁼×

This is a triadic, full program.

Try it online!

How it works

æ.⁵⁼×  Main link
       Left argument:  v (eigenvector)
       Right argument: λ (eigenvalue)
       Third argument: A (matrix)

  ⁵    Third; yield A.
æ.     Take the dot product of v and A, yielding Av.
    ×  Multiply v and λ component by component, yielding λv.
   ⁼   Test the results to the left and to the right for equality.