The GATE Grind

GATE 2023 EC – Question 15

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\{\lambda_k\mid1\le k\le n\}$ and $\{\boldsymbol v_k\mid1\le k\le n\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1}AP$, respectively, are

  1. $\{\lambda_k\det(A)\mid1\le k\le n\}$ and $\{P\boldsymbol v_k\mid1\le k\le n\}$
  2. $\{\lambda_k\mid1\le k\le n\}$ and $\{\boldsymbol v_k\mid1\le k\le n\}$
  3. $\{\lambda_k\mid1\le k\le n\}$ and $\{P\boldsymbol v_k\mid1\le k\le n\}$
  4. $\{\lambda_k\mid1\le k\le n\}$ and $\{P^{-1}\boldsymbol v_k\mid1\le k\le n\}$

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Correct answer: (C) $\{\lambda_k\mid1\le k\le n\}$ and $\{P\boldsymbol v_k\mid1\le k\le n\}$

Explanation

Similar matrices have the same eigenvalues. From $B\boldsymbol v_k=\lambda_k\boldsymbol v_k$ and $A=PBP^{-1}$: $A(P\boldsymbol v_k)=PB\boldsymbol v_k=\lambda_k(P\boldsymbol v_k)$. So the eigenvalues are unchanged and the eigenvectors of $A$ are $P\boldsymbol v_k$.