Given 2x2 rank-1 matrix A = xy^T where the dot product of x and y = 3

Question

Consider a 2 ×2 rank-1 matrix A= ⃗u⃗vT . Suppose you somehow know that ⃗u ·⃗v = 3; find the eigenvalues, the determinant and the trace of A.

I am confused about how I am supposed to find the elements of A given only the dot product of two vectors that form A when multiplied together.

Answer 1

Hints:

• An invertible $2\times 2$ matrix has rank $2$. What does this tell you about the determinant of a rank $1$ matrix?
• What is the relationship between the determinant of a matrix and its eigenvalues? From the determinant alone, you can deduce one of the eigenvalues.
• Write the trace of $A$ in terms of the entries of $x$ and $y$, noting that
  $$ A = xy^T = \pmatrix{x_1y_1 & x_1y_2\\x_2y_1&x_2y_2}. $$ How does this result relate to the dot-product of $x$ and $y$?
• What is the relationship between the trace of a matrix and its eigenvalues? Using the trace and the eigenvalue that you found in the previous step, find the other eigenvalue.