Let A = \(\left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right)\) and A = P + Q where P is symmetric matrix and Q is skew-symmetric matrix.
What is Q equal to ?
cos 2θ \(\left(\begin{array}{ccc}0 & −1/2 & 1/2 \\ 1/2 & 0 & −1/2 \\ −1/2 & 1/2 & 0\end{array}\right)\)
Understanding Matrix Decomposition
Any square matrix \(A\) can be uniquely expressed as the sum of a symmetric matrix \(P\) and a skew-symmetric matrix \(Q\). This is known as matrix decomposition. The formula for finding the symmetric part \(P\) and the skew-symmetric part \(Q\) are:
- Symmetric part \(P = \frac{1}{2}(A + A^T)\)
- Skew-symmetric part \(Q = \frac{1}{2}(A - A^T)\)
where \(A^T\) is the transpose of matrix \(A\).
What is a Symmetric Matrix?
A square matrix \(P\) is symmetric if it is equal to its transpose, i.e., \(P^T = P\). This means the elements \(p_{ij}\) are equal to \(p_{ji}\) for all \(i\) and \(j\).
What is a Skew-Symmetric Matrix?
A square matrix \(Q\) is skew-symmetric if its transpose is equal to the negative of the matrix, i.e., \(Q^T = -Q\). This means the elements \(q_{ij}\) are equal to \(-q_{ji}\) for all \(i\) and \(j\). A key property of skew-symmetric matrices is that their diagonal elements must be zero.
Step-by-Step Calculation to Find Q
We are given the matrix \(A\) and told that \(A = P + Q\), where \(P\) is symmetric and \(Q\) is skew-symmetric. We need to find the matrix \(Q\).
Given Matrix A
\(A = \left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right)\)
Finding the Transpose of A (\(A^T\))
The transpose of a matrix is obtained by interchanging its rows and columns.
\(A^T = \left(\begin{array}{ccc}0 & \cos^2 \theta & \sin^2 \theta \\ \sin^2 \theta & 0 & \cos^2 \theta \\ \cos^2 \theta & \sin^2 \theta & 0\end{array}\right)\)
Calculating A - A^T
Now, we subtract \(A^T\) from \(A\):
\(A - A^T = \left(\begin{array}{ccc}0 & \sin^2 \theta & \cos^2 \theta \\ \cos^2 \theta & 0 & \sin^2 \theta \\ \sin^2 \theta & \cos^2 \theta & 0\end{array}\right) - \left(\begin{array}{ccc}0 & \cos^2 \theta & \sin^2 \theta \\ \sin^2 \theta & 0 & \cos^2 \theta \\ \cos^2 \theta & \sin^2 \theta & 0\end{array}\right)\)
\(A - A^T = \left(\begin{array}{ccc}0-0 & \sin^2 \theta - \cos^2 \theta & \cos^2 \theta - \sin^2 \theta \\ \cos^2 \theta - \sin^2 \theta & 0-0 & \sin^2 \theta - \cos^2 \theta \\ \sin^2 \theta - \cos^2 \theta & \cos^2 \theta - \sin^2 \theta & 0-0\end{array}\right)\)
Using the trigonometric identity \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), we can simplify the off-diagonal elements:
- \(\sin^2 \theta - \cos^2 \theta = -(\cos^2 \theta - \sin^2 \theta) = -\cos 2\theta\)
- \(\cos^2 \theta - \sin^2 \theta = \cos 2\theta\)
So, \(A - A^T\) becomes:
\(A - A^T = \left(\begin{array}{ccc}0 & -\cos 2\theta & \cos 2\theta \\ \cos 2\theta & 0 & -\cos 2\theta \\ -\cos 2\theta & \cos 2\theta & 0\end{array}\right)\)
Calculating Q = 1/2(A - A^T)
Finally, we find the skew-symmetric matrix \(Q\) by multiplying the result by \(1/2\):
\(Q = \frac{1}{2} \left(\begin{array}{ccc}0 & -\cos 2\theta & \cos 2\theta \\ \cos 2\theta & 0 & -\cos 2\theta \\ -\cos 2\theta & \cos 2\theta & 0\end{array}\right)\)
\(Q = \left(\begin{array}{ccc}\frac{1}{2} \cdot 0 & \frac{1}{2} \cdot (-\cos 2\theta) & \frac{1}{2} \cdot (\cos 2\theta) \\ \frac{1}{2} \cdot (\cos 2\theta) & \frac{1}{2} \cdot 0 & \frac{1}{2} \cdot (-\cos 2\theta) \\ \frac{1}{2} \cdot (-\cos 2\theta) & \frac{1}{2} \cdot (\cos 2\theta) & \frac{1}{2} \cdot 0\end{array}\right)\)
\(Q = \left(\begin{array}{ccc}0 & -\frac{1}{2}\cos 2\theta & \frac{1}{2}\cos 2\theta \\ \frac{1}{2}\cos 2\theta & 0 & -\frac{1}{2}\cos 2\theta \\ -\frac{1}{2}\cos 2\theta & \frac{1}{2}\cos 2\theta & 0\end{array}\right)\)
We can factor out the common term \(\cos 2\theta\) from the matrix:
\(Q = \cos 2\theta \left(\begin{array}{ccc}0 & -\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & 0\end{array}\right)\)
Comparing with Options
Comparing our final calculated matrix \(Q\) with the given options, we see that it matches the form presented in Option 4.
Conclusion
The skew-symmetric matrix \(Q\) obtained from the decomposition of matrix \(A\) is:
\(Q = \cos 2\theta \left(\begin{array}{ccc}0 & -\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & 0\end{array}\right)\)
Revision Table: Matrix Decomposition Concepts
| Concept | Description | Property |
|---|---|---|
| Matrix Transpose (\(A^T\)) | Matrix obtained by interchanging rows and columns of A. | \((A^T)^T = A\), \((A+B)^T = A^T + B^T\), \((kA)^T = kA^T\), \((AB)^T = B^T A^T\) |
| Symmetric Matrix (P) | A square matrix equal to its transpose. | \(P^T = P\), \(p_{ij} = p_{ji}\) |
| Skew-Symmetric Matrix (Q) | A square matrix whose transpose is its negative. | \(Q^T = -Q\), \(q_{ij} = -q_{ji}\), \(q_{ii} = 0\) for all i. |
| Matrix Decomposition | Expressing a square matrix A as sum of a symmetric matrix P and a skew-symmetric matrix Q. | \(A = P + Q\), where \(P = \frac{1}{2}(A + A^T)\) and \(Q = \frac{1}{2}(A - A^T)\) |
Additional Information: Properties of Skew-Symmetric Matrices
- All diagonal elements of a skew-symmetric matrix are zero. This is because \(q_{ii} = -q_{ii}\) implies \(2q_{ii} = 0\), so \(q_{ii} = 0\).
- If \(A\) is a skew-symmetric matrix of odd order (e.g., 3x3, 5x5), its determinant is always zero, i.e., \(|A| = 0\).
- If \(A\) is a skew-symmetric matrix of even order (e.g., 2x2, 4x4), its determinant is a perfect square, i.e., \(|A| = (Pf(A))^2\), where \(Pf(A)\) is the Pfaffian of the matrix.
- The sum of two skew-symmetric matrices is a skew-symmetric matrix.
- Multiplying a skew-symmetric matrix by a scalar results in a skew-symmetric matrix.
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