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  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ be an $n \\times n$ matrix over $F$.\nThe *spectrum* of $A$ is the set\n$$\\sigma(A) = \\{\\lambda \\in F : A - \\lambda I_n \\text{ is not invertible}\\}.$$\nIt is the set of [[eigenvalues|0BQAA2BK3Q]] of $A$.\n\n## Remarks\n\nThe spectrum has at most $n$ elements, because each eigenvalue is a root of the characteristic polynomial of degree $n$.\nOver $\\C$, the spectrum of each square matrix is nonempty.\nThe spectrum depends on the field: the real matrix $\\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$ has empty spectrum over $\\R$ and spectrum $\\{i, -i\\}$ over $\\C$.\nSome authors count the eigenvalues with their multiplicities and call this list the spectrum.\n\n## Explanations\n\n### L1 human\n\nThe spectrum is the list of all eigenvalues of a matrix.\nIt tells us the scaling factors that the matrix shows in special directions.\nFor a diagonal matrix, it is the set of diagonal entries.\n",
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