Notes on Vector and Matrix Norms

Notes on Vector and Matrix NormsRobert A. van de GeijnDepartment of Computer ScienceThe University of Texas at AustinAustin, TX 15, 20141 Absolute ValueRecall that if C, then| |equals its absolute value. In other words, if = r+i c, then| |= 2r+ 2c= .This absolute value function has the following properties: 6= 0 | |>0 (| |is positive definite), | |=| || |(| |is homogeneous), and | + | | |+| |(| |obeys the triangle inequality).2 Vector NormsA ( Vector ) norm extends the notion of an absolute value (length or size) to vectors:Definition :Cn R. Then is a ( Vector ) norm if for allx,y Cn x6= 0 (x)>0( is positive definite), ( x) =| | (x)( is homogeneous), and (x+y) (x) + (y)( obeys the triangle inequality).

3 Matrix Norms It is not hard to see that vector norms are all measures of how \big" the vectors are. Similarly, we want to have measures for how \big" matrices are. We will start with one that are somewhat arti cial and then move on to the important class of induced matrix norms. 3.1 Frobenius norm De nition 12. The Frobenius norm kk F: Cm n!R ...

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