Prove as follows the inequality |Ax| =||A|| - |x|, where A is an m x m matrix with row vectors a1, a2, ..., am, and x is an m-dimensional vector. First note that the components of the vector Ax are aj · X, a2 • X, . .. , am • X, so 1/2 |Ax| = Σα2 Σ(a - x2 Then use the Cauchy-Schwarz inequality (a - x)? Ja2|x|? for the dot product. VI

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Prove as follows the inequality |Ax| =||A|| - |x|, where A is
an m x m matrix with row vectors a1, a2, ..., am, and x is
an m-dimensional vector. First note that the components
of the vector Ax are aj · X, a2 • X,
. .. , am • X, so
1/2
|Ax| =
Σα2
Σ(a - x2
Then use the Cauchy-Schwarz inequality (a - x)?
Ja2|x|? for the dot product.
VI
Transcribed Image Text:Prove as follows the inequality |Ax| =||A|| - |x|, where A is an m x m matrix with row vectors a1, a2, ..., am, and x is an m-dimensional vector. First note that the components of the vector Ax are aj · X, a2 • X, . .. , am • X, so 1/2 |Ax| = Σα2 Σ(a - x2 Then use the Cauchy-Schwarz inequality (a - x)? Ja2|x|? for the dot product. VI
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