Let A be a given square matrix. (a) Explain in detail why...

Let $A$ be a given square matrix. (a) Explain in detail why every nonzero scalar multiple of an eigenvector of $A$ is also an eigenvector. (b) Show that every nonzero linear combination of two eigenvectors $\mathbf{v}, \mathbf{w}$ corresponding to the same eigenvalue is also an eigenvector. (c) Prove that a linear combination $c \mathbf{v}+d \mathbf{w}$, with $c, d \neq 0$, of two eigenvectors corresponding to different eigenvalues is never an eigenvector.

Key Concepts

Subspace of Eigenvectors (Eigenspace)

The set of all eigenvectors corresponding to a specific eigenvalue, along with the zero vector, forms a subspace known as the eigenspace. This means that any nonzero linear combination of eigenvectors from the same eigenspace is itself an eigenvector with that eigenvalue. The closure of this set under addition and scalar multiplication is what grants this property.

Linear Combination of Eigenvectors from Distinct Eigenspaces

When forming a linear combination of eigenvectors that correspond to different eigenvalues, the resulting vector is not an eigenvector. This is because the transformation of each component leads to different scaling factors, making it impossible for a single scalar factor to emerge uniformly. This concept demonstrates the significance of eigenvalue multiplicity and separation in the analysis of linear transformations.

Eigenvector and Eigenvalue

Eigenvectors are nonzero vectors that, when transformed by a square matrix, result in a scalar multiple of themselves; this scalar is known as the eigenvalue. This concept is fundamental in analyzing how linear transformations behave, as it provides insight into invariant directions and the extent to which the transformation stretches or compresses vectors along those directions.

Scalar Multiplication Property

One of the key properties of eigenvectors is that any nonzero scalar multiple of an eigenvector is also an eigenvector corresponding to the same eigenvalue. This works because scaling a vector by a nonzero constant does not change its direction, ensuring that the action of the matrix still yields a scalar multiple of the original vector, leaving the eigenvalue unchanged.

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