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Sagot :
To express matrix [tex]\( A \)[/tex] as the sum of a symmetric matrix and a skew-symmetric matrix, we need to follow these steps:
1. Identify the given matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} \][/tex]
2. Find the symmetric part of [tex]\( A \)[/tex]:
The symmetric part of a matrix [tex]\( A \)[/tex] can be given by [tex]\(\frac{A + A^T}{2}\)[/tex], where [tex]\( A^T \)[/tex] is the transpose of [tex]\( A \)[/tex].
[tex]\[ A^T = \begin{pmatrix} 2 & 6 & 4 \\ 4 & 7 & 6 \\ 8 & 2 & 9 \end{pmatrix} \][/tex]
Now, add [tex]\( A \)[/tex] and [tex]\( A^T \)[/tex]:
[tex]\[ A + A^T = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} + \begin{pmatrix} 2 & 6 & 4 \\ 4 & 7 & 6 \\ 8 & 2 & 9 \end{pmatrix} = \begin{pmatrix} 4 & 10 & 12 \\ 10 & 14 & 8 \\ 12 & 8 & 18 \end{pmatrix} \][/tex]
Then, divide by 2:
[tex]\[ A_{sym} = \frac{A + A^T}{2} = \frac{1}{2} \begin{pmatrix} 4 & 10 & 12 \\ 10 & 14 & 8 \\ 12 & 8 & 18 \end{pmatrix} = \begin{pmatrix} 2 & 5 & 6 \\ 5 & 7 & 4 \\ 6 & 4 & 9 \end{pmatrix} \][/tex]
3. Find the skew-symmetric part of [tex]\( A \)[/tex]:
The skew-symmetric part of a matrix [tex]\( A \)[/tex] can be given by [tex]\(\frac{A - A^T}{2}\)[/tex].
[tex]\[ A - A^T = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} - \begin{pmatrix} 2 & 6 & 4 \\ 4 & 7 & 6 \\ 8 & 2 & 9 \end{pmatrix} = \begin{pmatrix} 0 & -2 & 4 \\ 2 & 0 & -4 \\ -4 & 4 & 0 \end{pmatrix} \][/tex]
Then, divide by 2:
[tex]\[ A_{skew} = \frac{A - A^T}{2} = \frac{1}{2} \begin{pmatrix} 0 & -2 & 4 \\ 2 & 0 & -4 \\ -4 & 4 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -2 \\ -2 & 2 & 0 \end{pmatrix} \][/tex]
4. Verify the decomposition:
To verify, we need to make sure [tex]\( A \)[/tex] equals the sum of the symmetric and skew-symmetric parts:
[tex]\[ A = A_{sym} + A_{skew} = \begin{pmatrix} 2 & 5 & 6 \\ 5 & 7 & 4 \\ 6 & 4 & 9 \end{pmatrix} + \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -2 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} \][/tex]
Thus, matrix [tex]\( A \)[/tex] can be expressed as the sum of its symmetric part:
[tex]\[ A_{sym} = \begin{pmatrix} 2 & 5 & 6 \\ 5 & 7 & 4 \\ 6 & 4 & 9 \end{pmatrix} \][/tex]
and its skew-symmetric part:
[tex]\[ A_{skew} = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -2 \\ -2 & 2 & 0 \end{pmatrix} \][/tex]
1. Identify the given matrix [tex]\( A \)[/tex]:
[tex]\[ A = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} \][/tex]
2. Find the symmetric part of [tex]\( A \)[/tex]:
The symmetric part of a matrix [tex]\( A \)[/tex] can be given by [tex]\(\frac{A + A^T}{2}\)[/tex], where [tex]\( A^T \)[/tex] is the transpose of [tex]\( A \)[/tex].
[tex]\[ A^T = \begin{pmatrix} 2 & 6 & 4 \\ 4 & 7 & 6 \\ 8 & 2 & 9 \end{pmatrix} \][/tex]
Now, add [tex]\( A \)[/tex] and [tex]\( A^T \)[/tex]:
[tex]\[ A + A^T = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} + \begin{pmatrix} 2 & 6 & 4 \\ 4 & 7 & 6 \\ 8 & 2 & 9 \end{pmatrix} = \begin{pmatrix} 4 & 10 & 12 \\ 10 & 14 & 8 \\ 12 & 8 & 18 \end{pmatrix} \][/tex]
Then, divide by 2:
[tex]\[ A_{sym} = \frac{A + A^T}{2} = \frac{1}{2} \begin{pmatrix} 4 & 10 & 12 \\ 10 & 14 & 8 \\ 12 & 8 & 18 \end{pmatrix} = \begin{pmatrix} 2 & 5 & 6 \\ 5 & 7 & 4 \\ 6 & 4 & 9 \end{pmatrix} \][/tex]
3. Find the skew-symmetric part of [tex]\( A \)[/tex]:
The skew-symmetric part of a matrix [tex]\( A \)[/tex] can be given by [tex]\(\frac{A - A^T}{2}\)[/tex].
[tex]\[ A - A^T = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} - \begin{pmatrix} 2 & 6 & 4 \\ 4 & 7 & 6 \\ 8 & 2 & 9 \end{pmatrix} = \begin{pmatrix} 0 & -2 & 4 \\ 2 & 0 & -4 \\ -4 & 4 & 0 \end{pmatrix} \][/tex]
Then, divide by 2:
[tex]\[ A_{skew} = \frac{A - A^T}{2} = \frac{1}{2} \begin{pmatrix} 0 & -2 & 4 \\ 2 & 0 & -4 \\ -4 & 4 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -2 \\ -2 & 2 & 0 \end{pmatrix} \][/tex]
4. Verify the decomposition:
To verify, we need to make sure [tex]\( A \)[/tex] equals the sum of the symmetric and skew-symmetric parts:
[tex]\[ A = A_{sym} + A_{skew} = \begin{pmatrix} 2 & 5 & 6 \\ 5 & 7 & 4 \\ 6 & 4 & 9 \end{pmatrix} + \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -2 \\ -2 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 4 & 8 \\ 6 & 7 & 2 \\ 4 & 6 & 9 \end{pmatrix} \][/tex]
Thus, matrix [tex]\( A \)[/tex] can be expressed as the sum of its symmetric part:
[tex]\[ A_{sym} = \begin{pmatrix} 2 & 5 & 6 \\ 5 & 7 & 4 \\ 6 & 4 & 9 \end{pmatrix} \][/tex]
and its skew-symmetric part:
[tex]\[ A_{skew} = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -2 \\ -2 & 2 & 0 \end{pmatrix} \][/tex]
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