／var／logmarcus chiu

Eigenspaces

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Eigenspaces
  • the eigenspace is the space generated by the eigenvectors corresponding to the same matrix - that is, the space of all vectors that can be written as a linear combination of those eigenvectors

𝜆-eigenspace

Let:

  • 𝐴 be an 𝑛×𝑛 matrix
  • 𝜆 be an eigenvalue of 𝐴

The 𝜆-eigenspace of 𝐴:

  • is the solution set of (𝐴−𝜆𝐼)𝑣=0
  • is the subspace 𝑁𝑢𝑙(𝐴−𝜆𝐼) i.e the null space of matrix 𝐴−𝜆𝐼
  • consists of the zero vector and all eigenvectors of 𝐴 with eigenvalue 𝜆
  • has a dimension equal to the number of free variables in the system of equations (𝐴−𝜆𝐼)𝑣=0, which is the number of columns of 𝐴−𝜆𝐼 without pivots

0-eigenspace

Fact

Let 𝐴 be an 𝑛×𝑛 matrix.

  1. The number 0 is an eigenvalue of 𝐴 if and only if 𝐴 is not invertible
  2. In this case, the 0-eigenspace of 𝐴 is 𝑁𝑢𝑙(𝐴)
Proof

We know that 0 is an eigenvalue of 𝐴 if and only if 𝑁𝑢𝑙(𝐴−0𝐼)=𝑁𝑢𝑙(𝐴) is nonzero, which is equivalent to the non-invertibility of 𝐴 by the invertible matrix theorem. In this case, the 0-eigenspace is by definition 𝑁𝑢𝑙(𝐴−0𝐼)=𝑁𝑢𝑙(𝐴).

Concretely, an eigenvector with eigenvalue 0 is a nonzero vector 𝑣 such that 𝐴𝑣=0𝑣, i.e., such that 𝐴𝑣=0. These are exactly the nonzero vectors in the null space of 𝐴