In linear algebra, what is an eigenvalue?

• A. The determinant of A.
• B. The trace of A.
• C. The inverse of A.
• D. A scalar λ such that AX = λX.

Answer: (D)

An eigenvalue is a scalar λ for which there exists a nonzero vector x such that A x = λ x. In other words, along the direction of x, the linear transformation A only stretches by a factor λ (or compresses if |λ| < 1) and does not change the direction of x. The vector x is called an eigenvector corresponding to that eigenvalue.

This concept is tied to how a matrix acts: some directions are preserved under the transformation, just scaled. For example, a diagonal matrix with entries 2 and 3 has eigenvalues 2 and 3 with eigenvectors along the coordinate axes, because applying the matrix to those basis vectors simply scales them.

Notes on the other ideas: the determinant is the product of all eigenvalues (counting multiplicities), not a single eigenvalue; the trace is the sum of all eigenvalues, not an eigenvalue itself; and if you invert A (when possible), the eigenvalues become the reciprocals 1/λ of the original ones (for nonzero λ).