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Chapter 6: Problem 6

Show that the diagonal entries of a Hermitian matrix must be real.

Short Answer

Expert verified
To show that the diagonal entries of a Hermitian matrix must be real, we start by considering an n脳n Hermitian matrix A, which means A = A*. The diagonal entries of A (denoted as \(a_{ii}\)) must also be equal to the diagonal entries of A* (denoted as \(\overline{a_{ii}}\)). The equality \(a_{ii} = \overline{a_{ii}}\) holds if and only if \(a_{ii}\) is real. Since the diagonal entries of A are equal to the diagonal entries of A* and this equality holds when the diagonal entries are real, we can conclude that the diagonal entries of a Hermitian matrix must be real.

Step by step solution

01

Definition of a Hermitian matrix

A complex square matrix A is Hermitian if it is equal to its Hermitian conjugate, denoted A*. The Hermitian conjugate of A is obtained by taking the transpose of the matrix and replacing each entry with its complex conjugate. In mathematical terms, A = A*.
02

Consider a Hermitian matrix A

Let A be an n脳n Hermitian matrix. Since A is equal to its Hermitian conjugate A*, the diagonal entries of A must also be equal to the diagonal entries of A*.
03

Calculate the diagonal entries of A*

To calculate the diagonal entries of A*, we need to find the complex conjugate of each diagonal entry of the transposed matrix. Since the diagonal entries do not change when taking the transpose of the matrix, we simply take the complex conjugate of each diagonal entry of A, denoted as \(\overline{a_{ii}}\).
04

Compare the diagonal entries of A and A*

Now, we compare the diagonal entries of A (denoted as \(a_{ii}\)) and A* (denoted as \(\overline{a_{ii}}\)). According to the definition of a Hermitian matrix, A = A*. Therefore, the diagonal entries of A and A* must be equal: \(a_{ii} = \overline{a_{ii}}\)
05

Determine the condition when the equality holds

The equality \(a_{ii} = \overline{a_{ii}}\) holds if and only if \(a_{ii}\) is real. To verify this, consider two cases: 1. If \(a_{ii}\) is real, then \(a_{ii} = \overline{a_{ii}}\) because the complex conjugate of a real number is the same real number. 2. If \(a_{ii}\) is not real, then \(a_{ii} \neq \overline{a_{ii}}\) because the complex conjugate of a non-real number is different from the original number.
06

Conclusion

Since the diagonal entries of A are equal to the diagonal entries of A* (due to A being Hermitian), and the equality holds if and only if the diagonal entries are real, we can conclude that the diagonal entries of a Hermitian matrix must be real.

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