If a matrix has eigenvalues, what do they represent in relation to the matrix transformation?

If a matrix has eigenvalues, what do they represent in relation to the matrix transformation?

• A. The factors by which the transformation scales corresponding eigenvectors
• B. The rotation angles of the transformation
• C. The number of fixed points of the transformation
• D. The rank of the matrix

Answer: (A)

Eigenvalues tell you how the matrix scales vectors that lie along directions which remain in the same direction after the transformation. If Ax = λx for a nonzero x, the transformation simply stretches or squeezes x by a factor λ (and may flip its direction if λ is negative). So the eigenvalue is the scaling factor for that eigenvector direction, which is exactly what the correct answer describes. Rotation angles aren’t given by eigenvalues—the eigenstructure doesn’t generally encode how much a transformation rotates space. The number of fixed points isn’t what eigenvalues measure, and rank concerns how many independent columns there are, not how the transformation scales specific directions.

Eigenvalues tell you how the matrix scales vectors that lie along directions which remain in the same direction after the transformation. If Ax = λx for a nonzero x, the transformation simply stretches or squeezes x by a factor λ (and may flip its direction if λ is negative). So the eigenvalue is the scaling factor for that eigenvector direction, which is exactly what the correct answer describes.

Rotation angles aren’t given by eigenvalues—the eigenstructure doesn’t generally encode how much a transformation rotates space. The number of fixed points isn’t what eigenvalues measure, and rank concerns how many independent columns there are, not how the transformation scales specific directions.