On a math note, if I understand you, the "given" vectors are the row vectors of the matrix {{-3,2},{-15,8}}, that is the vectors {-3, 2} and {-15, 8}. I do not think there is any general relationship between the row vectors and eigenvectors of a matrix.

The first column of a matrix mat is the vector you get from the product mat.{1,0}. The second column is the result of mat.{0,1}. An eigenvector is a (nonzero) linear combination of the columns that happens, when multiplied by mat, to yield a scalar multiple of itself. (Of course, every product mat.{x,y} is a linear combination of the columns of mat, so the scalar-multiple property is key for eigenvectors.)

Mitchell,

in your code you have too many curly brackets; try

Graphics[{Red, Arrow[{{0, 0}, 3 {1, 3}}], Blue, Arrow[{{0, 0}, 2 {2, 5}}]}, Axes -> True, AxesLabel -> {"x", "y"}]

For an understanding of eigenvectors here is a simple example to play with: Red are the eigenvectors, green represents the input vector and blue the resulting output. The vectors of the vector field gives the direction of this output.

mat = {{1., -.5}, {-1, -.5}};
{ev, evec} = Eigensystem[mat];
Manipulate[
 VectorPlot[mat . {x, y}, {x, -3, 3}, {y, -3, 3}, 
  Epilog -> {Red, Arrow[ev[[1]] {{0, 0}, evec[[1]]}], 
    Arrow[ev[[2]] {{0, 0}, evec[[2]]}], 
    Arrow[-ev[[1]] {{0, 0}, evec[[1]]}], 
    Arrow[-ev[[2]] {{0, 0}, evec[[2]]}], Green, Arrow[{{0, 0}, vec}], 
    Blue, Arrow[{{0, 0}, mat . vec}]}, 
  PerformanceGoal -> "Quality"], {{vec, {1, 1}}, Locator}]

Does that help? Regards -- Henrik