Chapter 8, Problem 1E

The singular values of any diagonal matrix D are the absolute values of the diagonal entries of D.

To determine

To state: the given statement is true or false

Answer to Problem 1E

The given statement is true

Explanation of Solution

Given:

The singular values of any diagonal matrix D are the absolute values of the diagonal entries of D.

Calculation:

Eigen values of $D^{T}D$ are $a_1^2,a_2^2,\mathrm{........},a_n^2,$ where numbers $a_1,a_2,\mathrm{.......},a_n$ are diagonal entries of D.

From definition of singular values, we have:

  ${\sigma }_1=\sqrt{a_i^2}=\left|a_i\right|,fori=1,\mathrm{......},n$

Conclusion:

Therefore, the given statement is true