Question

Find a 3×3 matrix with exactly one (real) eigenvalue 2, such that the 2-eigenspace is a line.

Answer 1

To find a 3x3 matrix with exactly one eigenvalue of 2 and an eigenspace that is a line, you can start by considering a diagonal matrix that has 2 as one of its diagonal entries.

Step-by-step explanation:

To find a 3x3 matrix with exactly one eigenvalue of 2 and an eigenspace that is a line, we can start by considering a diagonal matrix with 2 as one of its diagonal entries. For example:

[ 2 0 0 ]

[ 0 3 0 ]

[ 0 0 4 ]

Since the eigenvalue 2 has to correspond to an eigenvector, we can choose a vector that lies in the 2-eigenspace such as [1 0 0]. To construct a 3x3 matrix with a 2-eigenspace as a line, we can fill in the remaining two vectors with vectors that are linearly independent of [1 0 0], such as [0 1 0] and [0 0 1]. Therefore, a suitable matrix would be:

[ 2 0 0 ]

[ 0 3 0 ]

[ 0 0 4 ]

Answer 2

A 3×3 matrix with exactly one real eigenvalue (2) and an eigenspace that is a line can be constructed as a Jordan block, which allows for one eigenvalue to have a singular line as its eigenspace.

Step-by-step explanation:

To find a 3×3 matrix with only one real eigenvalue (2) and an eigenspace that is a line, we can consider that the eigenvalue 2 will have an algebraic multiplicity of 3 (since it is the only eigenvalue) but a geometric multiplicity of 1, which means that there is only one linearly independent eigenvector associated with it. A matrix with these properties could have the form of a Jordan block:

[2 1 0]

[0 2 1]

[0 0 2]

This matrix has the eigenvalue 2 with algebraic multiplicity 3, but the eigenvectors corresponding to the eigenvalue form only a line, which is the eigenspace. The Jordan block ensures that there is one linearly independent eigenvector but the other vectors related to the eigenvalue are generalized eigenvectors.