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Chapter 2: Problem 77

Suppose \(A\) is a complex matrix. Show that \(A A^{H}\) and \(A^{H} A\) are Hermitian.

Short Answer

Expert verified
We showed that \(AA^H\) and \(A^H A\) are both Hermitian by using properties of conjugate transpose and matrix multiplication. Specifically, we demonstrated that \((AA^H)^H = AA^H\) and \((A^H A)^H = A^H A\), which confirms that both products are Hermitian matrices as they are equal to their conjugate transpose.

Step by step solution

01

Compute the conjugate transpose of AA^H

First, let's find the conjugate transpose of the matrix product AA^H. To do this, we must use the properties of conjugate transpose and matrix multiplication: (AA^H)^H = (A^H)^H A^H
02

Simplify (A^H)^H

Recall the property that taking the conjugate transpose twice will give you the original matrix back: (A^H)^H = A
03

Substitute back

Now we can substitute Step 2 back into the result from Step 1: (AA^H)^H = AA^H As we have shown that the conjugate transpose of AA^H is equal to AA^H itself, it follows that AA^H is Hermitian.
04

Compute the conjugate transpose of A^H A

Next, let's find the conjugate transpose of the matrix product A^H A. We will use the same properties as in Step 1: (A^H A)^H = AH A^H
05

Simplify substituting A^H

Recall from step 2 that the conjugate transpose of A^H is the original matrix A. Substitute this into the result from Step 4: (A^H A)^H = A^H A As the conjugate transpose of A^H A is equal to A^H A itself, it follows that A^H A is also Hermitian. Since we have proven that AA^H and A^H A are both Hermitian matrices, the given exercise is complete.

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