True or false: (a) If λis an eigenvalue of both A and B,...

True or false: (a) If $\lambda$ is an eigenvalue of both $A$ and $B$, then it is an eigenvalue of the sum $A+B$. (b) If $\mathrm{v}$ is an eigenvector of both $A$ and $B$, then it is an eigenvector of $A+B$.

Key Concepts

Shared Eigenvectors and Simultaneous Diagonalizability

If a vector is an eigenvector for two matrices, then under the appropriate conditions (often when the matrices commute), it will be an eigenvector of their sum with the corresponding eigenvalue being the sum of the individual eigenvalues. This property is a key aspect in studying simultaneously diagonalizable matrices, although simply sharing an eigenvalue between matrices does not imply that the associated eigenvector is common to both.

Eigenvector

An eigenvector is a nonzero vector that changes at most by a scalar factor when a linear transformation is applied to it. It highlights directions in which the transformation acts merely as a scaling operation, making it essential for simplifying the representation of the matrix.

Eigenvalue

An eigenvalue is a scalar associated with a matrix such that there is a nonzero vector (called an eigenvector) which, when multiplied by the matrix, results in the scalar times that vector. This concept is central in understanding the scaling properties of linear transformations.

Matrix Addition and Eigenstructure

The process of adding two matrices involves adding their corresponding entries, but eigenvalues and eigenvectors do not add in a straightforward manner. In general, an eigenvalue of each individual matrix is not necessarily an eigenvalue of their sum, because the eigenstructure of a matrix is determined by its overall composition and interactions of its components.

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