Frobenius norm: Difference between revisions

Revision as of 02:45, 13 May 2014

Definition

For a matrix with real entries

Suppose $m,n$ are positive integers and $A$ is a $m\times n$ matrix. The Frobenius norm of $A$ , denoted $\|A\|_{F}$ , can be defined in the following equivalent ways:

It is the square root of the sum of squares of all the entries of $A$ , i.e., it is the sum ${\sqrt {\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}^{2}}}$ .
It is the square root of the trace of the $m\times m$ matrix $AA^{T}$ , where $A^{T}$ is the matrix transpose of $A$ .
It is the square root of the trace of the $n\times n$ matrix $A^{T}A$ , where $A^{T}$ is the matrix transpose of $A$ .
It is the root mean square of the $\min\{m,n\}$ many singular values of $A$ .

The Frobenius norm is invariant under orthogonal transformations (and in particular, under rotations) and is an easy-to-compute invariant.

For a matrix with complex entries

Suppose $m,n$ are positive integers and $A$ is a $m\times n$ matrix. The Frobenius norm of $A$ , denoted $\|A\|_{F}$ , can be defined in the following equivalent ways:

It is the square root of the sum of squares of the moduli of the entries of $A$ , i.e., it is the sum $\sum _{i=1}^{m}\sum _{j=1}^{n}|a_{ij}|^{2}$ .
It is the square root of the trace of the matrix $AA^{*}$ where $A^{*}$ is the matrix conjugate transpose of $A$ .
It is the square root of the trace of the matrix $A^{*}A$ where $A^{*}$ is the matrix conjugate transpose of $A$ .
It is the root mean square of the $\min\{m,n\}$ many singular values of $A$ .

@@ Line 5: / Line 5: @@
 Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:
-# It is the sum of squares of all the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2</math>.
+# It is the square root of the sum of squares of all the entries of <math>A</math>, i.e., it is the sum <math>\sqrt{\sum_{i=1}^m \sum_{j=1}^n a_{ij}^2}</math>.
-# It is the trace of the <math>m \times m</math> matrix <math>AA^T</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
+# It is the square root of the trace of the <math>m \times m</math> matrix <math>AA^T</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
-# It is the trace of the <math>n \times n</math> matrix <math>A^TA</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
+# It is the square root of the trace of the <math>n \times n</math> matrix <math>A^TA</math>, where <math>A^T</math> is the [[matrix transpose]] of <math>A</math>.
 # It is the [[root mean square]] of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.
@@ Line 16: / Line 16: @@
 Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''Frobenius norm''' of <math>A</math>, denoted <math>\| A \|_F</math>, can be defined in the following equivalent ways:
-# It is the sum of squares of the moduli of the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2</math>.
+# It is the square root of the sum of squares of the moduli of the entries of <math>A</math>, i.e., it is the sum <math>\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2</math>.
-# It is the trace of the matrix <math>AA^*</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
+# It is the square root of the trace of the matrix <math>AA^*</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
-# It is the trace of the matrix <math>A^*A</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
+# It is the square root of the trace of the matrix <math>A^*A</math> where <math>A^*</math> is the [[matrix conjugate transpose]] of <math>A</math>.
 # It is the [[root mean square]] of the <math>\min \{m, n \}</math> many [[singular value]]s of <math>A</math>.