Editing Matrix norm (section)

===Frobenius norm===
{{Main|Hilbert–Schmidt operator}}
{{See also|Frobenius inner product}}

When {{nowrap|1=''p'' = ''q'' = 2}} for the <math>L_{p,q}</math> norm, it is called the '''Frobenius norm''' or the '''Hilbert–Schmidt norm''', though the latter term is used more frequently in the context of operators on (possibly infinite-dimensional) [[Hilbert space]]. This norm can be defined in various ways:

:<math>\|A\|_\text{F} = \sqrt{\sum_{i}^m\sum_{j}^n |a_{ij}|^2} = \sqrt{\operatorname{trace}\left(A^* A\right)} = \sqrt{\sum_{i=1}^{\min\{m, n\}} \sigma_i^2(A)},</math>

where the [[trace (matrix)|trace]] is the sum of diagonal entries, and <math>\sigma_i(A)</math> are the [[singular value]]s of <math>A</math>. The second equality is proven by explicit computation of <math>\mathrm{trace}(A^*A)</math>. The third equality is proven by [[singular value decomposition]] of <math>A</math>, and the fact that the trace is invariant under circular shifts.

The Frobenius norm is an extension of the Euclidean norm to <math>K^{n \times n}</math> and comes from the [[Frobenius inner product]] on the space of all matrices.

The Frobenius norm is sub-multiplicative and is very useful for [[numerical linear algebra]]. The sub-multiplicativity of Frobenius norm can be proved using the [[Cauchy–Schwarz inequality]]. In fact, it is more than sub-multiplicative, as <math display="block">\|AB\|_F \leq\|A\|_{op}\|B\|_F</math>where the operator norm <math>\|\cdot\|_{op} \leq \|\cdot\|_{F}</math>.

Frobenius norm is often easier to compute than induced norms, and has the useful property of being invariant under [[rotation matrix|rotations]] (and [[Unitary operator|unitary]] operations in general). That is, <math>\|A\|_\text{F} = \|AU\|_\text{F} = \|UA\|_\text{F}</math> for any unitary matrix <math>U</math>. This property follows from the cyclic nature of the trace (<math>\operatorname{trace}(XYZ) =\operatorname{trace}(YZX) = \operatorname{trace}(ZXY)</math>):

:<math>\|AU\|_\text{F}^2 = \operatorname{trace}\left( (AU)^{*}A U \right)
  = \operatorname{trace}\left( U^{*} A^{*}A U \right)
  = \operatorname{trace}\left( UU^{*} A^{*}A \right)
  = \operatorname{trace}\left( A^{*} A \right)
  = \|A\|_\text{F}^2,</math>

and analogously:

:<math>\|UA\|_\text{F}^2 = \operatorname{trace}\left( (UA)^{*}UA \right)
  = \operatorname{trace}\left( A^{*} U^{*} UA  \right)
  = \operatorname{trace}\left( A^{*}A \right)
  = \|A\|_\text{F}^2,</math>

where we have used the unitary nature of <math>U</math> (that is, <math>U^* U = U U^* = \mathbf{I}</math>).

It also satisfies

:<math>\|A^* A\|_\text{F} = \|AA^*\|_\text{F} \leq \|A\|_\text{F}^2</math>
and 
:<math>\|A + B\|_\text{F}^2 = \|A\|_\text{F}^2 + \|B\|_\text{F}^2 + 2 \operatorname{Re} \left( \langle A, B \rangle_\text{F} \right),</math>

where <math>\langle A, B \rangle_\text{F}</math> is the [[Frobenius inner product]], and Re is the real part of a complex number (irrelevant for real matrices)