Numerical Linear Algebra and Matrix Factorizations by Tom Lyche

Author:Tom Lyche
Language: eng
Format: epub, pdf
ISBN: 9783030364687
Publisher: Springer International Publishing

By Lemma 7.​1 we have , for all . Thus the Frobenius norm is subordinate to the Euclidian vector norm.

For consistent matrix norms we have

Proposition 8.1

For , , all and any consistent matrix norm

(8.11)

i.e., a consistent matrix norm is subordinate to itself. Moreover, the matrix power bound (8.10) holds for all square matrices .

Proof

Since a consistent matrix norm is defined on for all the consistency implies that (8.11) holds for and . The last statement also follows immediately from the consistency. □

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