Matrix Norm

Definition. Frobenius Norm

• \(A \in \mathbb{C}^{m \times n}\),

\[\begin{eqnarray} \|A\|_{\mathrm{F}} & := & \sqrt{ \sum_{i=1}^{m} \sum_{j=1}^{n} (a_{j}^{i})^{2} } \nonumber \\ & = & \sqrt{ \sum_{i=1}^{m} (a^{i})^{\mathrm{T}} (a^{i}) } \nonumber \\ & = & \sqrt{ \sum_{j=1}^{n} (a_{j})^{\mathrm{T}} (a_{j}) } \end{eqnarray}\]

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Proposition 2.

• \(A = (a_{1} \cdot a_{n}) \in \mathbb{C}^{m \times n}\),
  • \(a_{i} \in \mathbb{C}^{m}\) is column vectors of $A$.

\[\|A \|_{\mathrm{F}}^{2} = \mathrm{tr}(A^{\mathrm{T}}A) = \mathrm{tr}(AA^{\mathrm{T}})\]

proof.

We show first equality as follows:

\[\begin{eqnarray} \mathrm{tr}(A^{\mathrm{T}}A) & = & \left( \begin{array}{c} a_{1}^{\mathrm{T}} \\ \vdots \\ a_{n}^{\mathrm{T}} \end{array} \right) (a_{1} \cdot a_{n}) \\ & = & \mathrm{tr} \left( (a_{j_{1}}^{\mathrm{T}}a_{j_{2}})_{j_{1},j_{2}=1,\ldots, n} \right) \nonumber \\ & = & \sum_{j=1}^{n} (a_{j}^{\mathrm{T}}a_{j}) . \end{eqnarray}\] \[\begin{eqnarray} \mathrm{tr}(AA^{\mathrm{T}}) & = & \mathrm{tr}((A^{\mathrm{T}}A)^{\mathrm{T}}) \nonumber \\ & = & \mathrm{tr} \left( \left( (a_{j_{1}}^{\mathrm{T}}a_{j_{2}})_{j_{1},j_{2}=1,\ldots, n} \right)^{\mathrm{T}} \right) \nonumber \\ & = & \sum_{j=1}^{n} (a_{j}^{\mathrm{T}}a_{j}) \nonumber . \end{eqnarray}\]

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Proposition3.

• $A \in \mathbb{R}^{m \times n}$,
• $U \in \mathbb{R}^{m \times m}$
  • unitary matrix
• $V \in \mathbb{R}^{n \times n}$
  • unitary matrix

Then

\[\|UAV\|_{F} = \|A\|_{F} .\]

proof.

\[\begin{eqnarray} \|UA\|_{F}^{2} & = & \mathrm{tr} ( (UA)^{\mathrm{T}}UA ) \nonumber \\ & = & \mathrm{tr} ( A^{\mathrm{T}}A ) \nonumber \end{eqnarray}\]

Moreover,

\[\begin{eqnarray} \|AV\|_{F}^{2} & = & \mathrm{tr} ( AV(AV)^{\mathrm{T}} ) \nonumber \\ & = & \mathrm{tr} ( AA^{\mathrm{T}} ) . \nonumber \end{eqnarray}\]

$\Box$

Propositon 4.

• \(A \in \mathbb{C}^{m \times n}\),
• \(U \in \mathbb{C}^{m \times m}\),
  • unitary matrix
• \(\Sigma \in \mathbb{C}^{m \times n}\),
  • \(\Sigma = \mathrm{diag}(\sigma_{1}, \ldots, \sigma_{r}, 0, \ldots, 0)\),
  • singular value of $A$
• \(V \in \mathbb{C}^{n \times n}\),
  • unitary matrix

Suppose SVD $A = U\Sigma V$ holds. Then

\[\|A\|_{F}^{2} = \sum_{i=1}^{r} (\sigma_{i})^{2}\]

proof.

By the above proposition,

\[\begin{eqnarray} \|A\|_{F}^{2} & = & \|U\Sigma V\|_{F}^{2} \nonumber \\ & = & \|\Sigma \|_{F}^{2} \nonumber \\ & = & \mathrm{tr}(\Sigma \Sigma) \nonumber \\ & = & \mathrm{tr}(\Sigma^{2}) \nonumber \end{eqnarray}\]

$\Box$

Proposition5

• $S$
  • symmetric (i.e. $S = S^{\mathrm{T}}$),
• $K$
  • skew symmetric (i.e. $K = -K^{\mathrm{T}}$),

Then

\[\|S + K\|_{F}^{2} = \|S\|_{F}^{2} + \|K\|_{F}^{2}\]

proof.

\[\begin{eqnarray} \|S + T\|_{F}^{2} & = & \mathrm{tr}((S + T)^{\mathrm{T}}(S + T)) \mathrm{tr} \left( (S^{\mathrm{T}} + T^{\mathrm{T}}) (S + T) \right) \nonumber \\ & = & \mathrm{tr} \left( S^{\mathrm{T}}S + T^{\mathrm{T}}S + S^{\mathrm{T}}T + T^{\mathrm{T}}S \right) \nonumber \\ & = & \mathrm{tr} \left( S^{\mathrm{T}}S + (-TS) + (TS)^{\mathrm{T}} + T^{\mathrm{T}}T \right) \nonumber \\ & = & \mathrm{tr} \left( S^{\mathrm{T}}S \right) + \mathrm{tr} \left( -TS \right) + \mathrm{tr} \left( (TS)^{\mathrm{T}} \right) + \mathrm{tr} \left( T^{\mathrm{T}}T \right) \nonumber \\ & = & \|S\|_{F}^{2} + \|T\|_{F}^{2} \nonumber \end{eqnarray}\]

$\Box$

Reference

• 行列のフロベニウスノルムとその性質 高校数学の美しい物語