Page 94 - Matrix Analysis & Applied Linear Algebra

88 Chapter 3 Matrix Algebra

3.2.4. Explain why the set of all n × n symmetric matrices is closed under
matrix addition. That is, explain why the sum of two n × n symmetric
matrices is again an n × n symmetric matrix. Is the set of all n × n
skew-symmetric matrices closed under matrix addition?

3.2.5. Prove that each of the following statements is true.
(a) If A =[a ij ] is skew symmetric, then a jj = 0 for each j.
(b) If A =[a ij ] is skew hermitian, then each a jj is a pure imagi-
nary number—i.e., a multiple of the imaginary unit i.
(c) If A is real and symmetric, then B =iA is skew hermitian.

3.2.6. Let A be any square matrix.
T
T
(a) Show that A+A is symmetric and A−A is skew symmetric.
(b) Prove that there is one and only one way to write A as the
sum of a symmetric matrix and a skew-symmetric matrix.

3.2.7. If A and B are two matrices of the same shape, prove that each of the
following statements is true.
∗
(a) (A + B) = A + B .
∗
∗
∗
(b) (αA) = αA .
∗
3.2.8. Using the conventions given in Example 3.2.1, determine the stiffness
matrix for a system of n identical springs, with stiffness constant k,
connected in a line similar to that shown in Figure 3.2.1.