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For all matrices '''A''' and '''B''' in ''M''<sub>''mn''</sub>(''F''), and all numbers ''α'' in ''F'', a matrix norm, delimited by double vertical bars || ... ||, fulfills:<ref group="note">Some authors, e.g. Horn and Johnson, use triple vertical bars instead of double: |||'''A'''|||.</ref>
*[[
::<math>\| \mathbf{A} \| \ge 0</math>
:with equality only for '''A''' = '''0''', the [[zero matrix]].
*[[Scalar multiplication]]:
::<math>\|\alpha \mathbf{A}\|=|\alpha| \|\mathbf{A}\|</math>
*The [[triangular inequality]]:
::<math>\|\mathbf{A}+\mathbf{B}\| \leq \|\mathbf{A}\|+\|\mathbf{B}\|</math>
===Frobenius norm===
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