Question

Suppose a matrix A E RMXn has singular values n, n – 1, ...,1. Determine the value of the following maximum ratio XT AY max 07YER", || 2 || 2 ||4|| 2 OFxER 8. For a matrix A ERmxn, if the biggest singular value of A is 01 > 0, show that ||AT Y||2 0#YERM ||y||2 max = 01.

Answer 1

The value of the maximum ratio XTAYmaxOYER is n/1. The norm of ATY is less than or equal to the norm of Y.

Step-by-step explanation:

To determine the value of the maximum ratio XTAYmaxOYER, we need to consider the singular values of matrix A. The maximum ratio can be found by dividing the largest singular value of A by the smallest singular value of A, which is 1. Therefore, the value of the maximum ratio is n/1, where n is the number of columns in matrix A.

To show that ||ATY||2 <= ||y||2 max = 01, we need to use the properties of singular values. The largest singular value of A is greater than or equal to the Euclidean norm of ATY divided by the Euclidean norm of Y. Since the largest singular value is 01, we can conclude that ||ATY||2 <= ||y||2 max = 01.