Let a_1,a_2… be the eigenvalues of symmetric positive semidefinite matrix A. How to find the eigenvalues of matrix A(A+I)^(-1)?

kasibug1v 2022-10-02 Answered
Let a 1 , a 2 be the eigenvalues of symmetric positive semidefinite matrix A. How to find the eigenvalues of matrix A ( A + I ) 1 ?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Paige Paul
Answered 2022-10-03 Author has 11 answers
Let A = Q Λ Q T be the spectral decomposition. Then A + I = Q Λ Q T + Q Q T = Q ( Λ + I ) Q T . Thus, when we compute the inverse, we get that
A ( A + I ) 1 = Q Λ Q T ( Q ( Λ + 1 ) Q T ) 1 = Q Λ Q T Q ( Λ + I ) 1 Q T = Q Λ ( Λ + I ) 1 Q T .
Thus is λ i is an eigenvalue of A, we have that λ i λ i + 1 is an eigenvalue of A ( A + I ) 1
In general if all matrices are simultaneously diagonalizable then, we can just assume that our matrices are diagonal and derive the result for this case to get the general result.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions