Spectrum of a matrix

Definition

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$. The spectrum of $A$ is the set $$\sigma(A) = \{\lambda \in F : A - \lambda I_n \text{ is not invertible}\}.$$ It is the set of eigenvalues of $A$.

Remarks

The spectrum has at most $n$ elements, because each eigenvalue is a root of the characteristic polynomial of degree $n$. Over $\C$, the spectrum of each square matrix is nonempty. The spectrum depends on the field: the real matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ has empty spectrum over $\R$ and spectrum $\{i, -i\}$ over $\C$. Some authors count the eigenvalues with their multiplicities and call this list the spectrum.

Explanations

L1 human

The spectrum is the list of all eigenvalues of a matrix. It tells us the scaling factors that the matrix shows in special directions. For a diagonal matrix, it is the set of diagonal entries.