Join the conversation on IDNLearn.com and get the answers you seek from experts. Ask anything and receive well-informed answers from our community of experienced professionals.
Sagot :
To find the conjugate transpose (also known as the Hermitian transpose) of a matrix, you need to follow these steps:
1. Transpose the matrix: Swap rows with columns.
2. Conjugate each element: Replace each element with its complex conjugate.
Given matrix:
[tex]\[ \begin{pmatrix} 1 & 2+i \\ 5 & 5 \\ 9+7i & -3-3i \end{pmatrix} \][/tex]
### Step 1: Transpose the Matrix
The transpose of the matrix is obtained by swapping rows with columns:
[tex]\[ \begin{pmatrix} 1 & 5 & 9+7i \\ 2+i & 5 & -3-3i \end{pmatrix} \][/tex]
### Step 2: Conjugate Each Element
To take the complex conjugate of each element:
- For a real number [tex]\(a\)[/tex], the complex conjugate is the number itself, [tex]\(a\)[/tex].
- For a complex number [tex]\(a + bi\)[/tex], the complex conjugate is [tex]\(a - bi\)[/tex].
Applying this to the transposed matrix:
[tex]\[ \begin{pmatrix} 1 & 5 & 9+7i \\ 2+i & 5 & -3-3i \end{pmatrix} \][/tex]
1. [tex]\(1 \rightarrow 1 - 0i = 1\)[/tex]
2. [tex]\(5 \rightarrow 5 - 0i = 5\)[/tex]
3. [tex]\(9+7i \rightarrow 9-7i\)[/tex]
4. [tex]\(2+i \rightarrow 2-i\)[/tex]
5. [tex]\(5 \rightarrow 5 - 0i = 5\)[/tex]
6. [tex]\(-3-3i \rightarrow -3+3i\)[/tex]
Thus, the conjugated elements are:
[tex]\[ \begin{pmatrix} 1 & 5 & 9-7i \\ 2-i & 5 & -3+3i \end{pmatrix} \][/tex]
### Final Answer:
The conjugate transpose of the given matrix is:
[tex]\[ \begin{pmatrix} 1 & 5 & 9-7i \\ 2-i & 5 & -3+3i \end{pmatrix} \][/tex]
1. Transpose the matrix: Swap rows with columns.
2. Conjugate each element: Replace each element with its complex conjugate.
Given matrix:
[tex]\[ \begin{pmatrix} 1 & 2+i \\ 5 & 5 \\ 9+7i & -3-3i \end{pmatrix} \][/tex]
### Step 1: Transpose the Matrix
The transpose of the matrix is obtained by swapping rows with columns:
[tex]\[ \begin{pmatrix} 1 & 5 & 9+7i \\ 2+i & 5 & -3-3i \end{pmatrix} \][/tex]
### Step 2: Conjugate Each Element
To take the complex conjugate of each element:
- For a real number [tex]\(a\)[/tex], the complex conjugate is the number itself, [tex]\(a\)[/tex].
- For a complex number [tex]\(a + bi\)[/tex], the complex conjugate is [tex]\(a - bi\)[/tex].
Applying this to the transposed matrix:
[tex]\[ \begin{pmatrix} 1 & 5 & 9+7i \\ 2+i & 5 & -3-3i \end{pmatrix} \][/tex]
1. [tex]\(1 \rightarrow 1 - 0i = 1\)[/tex]
2. [tex]\(5 \rightarrow 5 - 0i = 5\)[/tex]
3. [tex]\(9+7i \rightarrow 9-7i\)[/tex]
4. [tex]\(2+i \rightarrow 2-i\)[/tex]
5. [tex]\(5 \rightarrow 5 - 0i = 5\)[/tex]
6. [tex]\(-3-3i \rightarrow -3+3i\)[/tex]
Thus, the conjugated elements are:
[tex]\[ \begin{pmatrix} 1 & 5 & 9-7i \\ 2-i & 5 & -3+3i \end{pmatrix} \][/tex]
### Final Answer:
The conjugate transpose of the given matrix is:
[tex]\[ \begin{pmatrix} 1 & 5 & 9-7i \\ 2-i & 5 & -3+3i \end{pmatrix} \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.