Question

Norm of matrix-vector product. Suppose A is an m !! n matrix and x is an n-vector. A famous inequality relates ||x||, ||A||, and ||Ax||: ||Ax|| ? ||A||||x||. The left-hand side is the (vector) norm of the matrix-vector product; the right-hand side is the (scalar) product of the matrix and vector norms. Show this inequality. Hints. Let a_i^T be the ith row of A. Use the Cauchy–Schwarz inequality to get (a_i^T x)^2 ? ||a_i||^2||x||^2. Then add the resulting m inequalities.

          Norm of matrix-vector product. Suppose A is an m !! n matrix and x is an n-vector. A famous inequality relates ||x||, ||A||, and ||Ax||:
||Ax|| ? ||A||||x||.
The left-hand side is the (vector) norm of the matrix-vector product; the right-hand side is the (scalar) product of the matrix and vector norms. Show this inequality. Hints. Let a_i^T be the ith row of A. Use the Cauchy–Schwarz inequality to get (a_i^T x)^2 ? ||a_i||^2||x||^2. Then add the resulting m inequalities.

Norm of matrix-vector product. Suppose A is an m !! n matrix and x is an n-vector. A famous inequality relates ||x||, ||A||, and ||Ax||:
||Ax|| ? ||A||||x||.
The left-hand side is the (vector) norm of the matrix-vector product; the right-hand side is the (scalar) product of the matrix and vector norms. Show this inequality. Hints. Let ai^T be the ith row of A. Use the Cauchy–Schwarz inequality to get (ai^T x)^2 ? ||ai||^2||x||^2. Then add the resulting m inequalities.

Added by Irene C.

Question

Please give Ace some feedback