Eigensystems - Basic

 

• x*x = 2
  

• x*x = 0
  

• xx* is a 2x2 real matrix
  

• xx* is a 2x2 Hermitian matrix
  

• rank(xx*) = 2
  

An nxn matrix has n eigenvalues counting multiplicities.

• True

• False

Let X be a matrix partitioned as X = [X₁ X₂] . Which of the following is correct?

• X^TX = X₁^TX₁ + X₂^TX₂

• The diagonal blocks of X^TX are X₁^TX₁ and X₂^TX_2.

Let p(λ) = det(λI - A) and q(λ) = det(A-λI). Do p(λ) and q(λ) have the same roots?

• Yes

• No

If a complex matrix has a real eigenvalue then the corresponding eigenvector can be chosen to have real entries. True or false?

• False in general.

• True.