Math 220 - April 23, 1999

Finding 1-dimensional Subspaces Invariant Under a Linear Map

• Recall that a 1-dimensional vector space is the set of all multiples of any of the non-zero vectors that lie in it. That is, any non-zero vector in a 1-dimensional subspace forms, by itself, a basis of that subspace.

• If L is a 1-dimensional subspace of V, i.e., if L is a line through the origin, and f is a linear map from V to itself, then L is f-invariant if and only if f(u) is in L for every u in L. This is true if and only if f(u) is in L for some non-zero vector u in L.

• Continuation. The following statements about f and L are equivalent:

  1. L is an f-invariant subspace.

  2. f(u_{0}) = lambda u_{0} for some scalar lambda and some non-zero vector u_{0} in L.

  3. f(u) = lambda u for every u in L and some scalar lambda that is independent of u.

• Continuation. The following statements about f, L, and a scalar lambda are equivalent:

  1. The matrix of ``f restricted to L'' is the 1 \times 1 matrix lambda in any basis of L.

  2. f agrees with the linear map ``multiplication by lambda'' on the subspace L.

  3. L is contained in the kernel of the linear map lambda . 1_{V} - f (1_{V} denotes the identity map of V).

  4. If M is the matrix of f relative to a basis of V, then the square matrix lambda 1 - M is not invertible, i.e., its determinant is zero, and its null space contains the coordinate columns of the vectors in L relative to the given basis of V.

• Definition. A scalar lambda that arises from a linear map or from a matrix in the preceding discussion is said to be a proper value of the linear map or of the matrix. Synonyms: Characteristic value or eigenvalue.

• Definition. If lambda is a proper value of a linear map f or of a matrix M as above, then the kernel of lambda . 1_{V} - f or the null space of the matrix lambda 1 - M is called the characteristic subspace of the linear map or of the matrix for the proper value lambda. Synonyms: Eigenspace or proper subspace.

Assignment for Monday, April 26

In the following exercises find all proper values and, for each proper value, the corresponding characteristic subspace of the linear map or of the matrix, as specified.

1. The 2 \times 2 matrix

   (

   1	-1
   0	1

   ) .

2. The rotation of R^{2} about the origin counterclockwise through a given angle theta, 0 < theta < pi/2.

3. The 3 \times 3 matrix

   {1}/{3}(

   1	2	2
   2	1	-2
   2	-2	1

   ) .

4. The mirror reflection of R^{3} in the plane x + 2 y + 2 z = 0 .