Eigenvalues

• Why Every Real Symmetric Matrix Is Diagonalizable
  A real symmetric matrix has an orthonormal basis of eigenvectors, so an orthogonal change of basis diagonalizes it.
• How Do Projection and Reflection Linear Transformations Work?
  Build projection and reflection matrices, compose them in the correct order, and check their eigenvalues with a worked two-dimensional example.
• What Are Left and Right Eigenvalues?
  For an ordinary real or complex square matrix, left and right eigenvalues are the same; the associated eigenvectors satisfy different equations.
• How to Calculate an Eigenspace
  Calculate an eigenspace by solving the homogeneous system (A − λI)x = 0, then express its solution set as a span or basis.
• Can One Eigenvalue Have Two Different Eigenvectors?
  Yes. An eigenvalue can have many eigenvectors: every nonzero vector in its eigenspace qualifies, including scalar multiples and, sometimes, independent vectors.