Does There Exist a 3x3 Skew-Symmetric and Orthogonal Matrix?

Mathematics often confronts us with intriguing questions that challenge our understanding of matrix properties. One such question revolves around the existence of a 3x3 matrix that is both skew-symmetric and orthogonal. This article provides a comprehensive exploration of why such a matrix cannot exist, combining theoretical insights with mathematical proofs.

Understanding Skew-Symmetric and Orthogonal Matrices

To comprehend why a 3x3 matrix cannot be both skew-symmetric and orthogonal, let's first define the terms:

Skew-Symmetric Matrix

A matrix (A) is skew-symmetric if it satisfies the condition (A^T -A). This means that for any (i) and (j), the entry (a_{ij}) is equal to the negative of the entry (a_{ji}).

Orthogonal Matrix

A matrix (A) is orthogonal if it satisfies the condition (A^T A I), where (I) is the identity matrix. This implies that the columns (and rows) of (A) form an orthonormal set, meaning each column (and row) is a unit vector and the dot product of any two distinct columns (rows) is zero.

Contradiction in the 3x3 Case

Let's assume that there exists a 3x3 matrix (A) that is both skew-symmetric and orthogonal. This matrix must satisfy both conditions simultaneously:

Skew-Symmetry: (A^T -A) Orthogonality: (A A^T I)

From the skew-symmetry condition, we know that the diagonal entries of (A) must be zero because (a_{ii} -a_{ii} Rightarrow a_{ii} 0). Therefore, the main diagonal of (A) would look like:

[ A begin{pmatrix} 0 a b -a 0 c -b -c 0 end{pmatrix} ]

Now, let's use the orthogonality condition. For the matrix (A) to be orthogonal, we need:

[ A A^T begin{pmatrix} 0 a b -a 0 c -b -c 0 end{pmatrix} begin{pmatrix} 0 -a -b a 0 -c b c 0 end{pmatrix} begin{pmatrix} -1 0 0 0 -1 0 0 0 -1 end{pmatrix} ]

This gives us the following relations:

Relation Analysis

By expanding the product, we obtain the following equations:

[ begin{aligned} a^2 b^2 1 a^2 c^2 1 b^2 c^2 1 ab 0 ac 0 bc 0 end{aligned} ]

The first set of equations ((a^2 b^2 1, a^2 c^2 1, b^2 c^2 1)) implies that each pair of the variables (a, b, c) must take the value (pm 1) or (0). However, the second set of equations ((ab 0, ac 0, bc 0)) means that at most one of (a, b, c) can be non-zero. This creates a contradiction because if any one of (a, b, c) is non-zero, the others must be zero, but this would violate the first set of equations.

Proof of the Determinant for Odd-Ordered Skew-Symmetric Matrices

To further solidify the impossibility, let's consider the proof for the determinant of an odd-ordered skew-symmetric matrix:

Lemma

The determinant of a (2k-1) x (2k-1) skew-symmetric matrix is zero.

Proof

Assume (M) is a ((2k-1)) x ((2k-1)) skew-symmetric matrix:

[ M^T -M ]

This gives:

[ text{det}(M^T) text{det}(-M) (-1)^{(2k-1)} cdot text{det}(M) ]

Since the determinant of a matrix is equal to the determinant of its transpose, we have:

[ text{det}(M) text{det}(M^T) ]

This results in:

[ text{det}(M) (-1)^{(2k-1)} cdot text{det}(M) ]

For this equality to hold, (text{det}(M)) must be zero. Therefore, the determinant of a (2k-1) x (2k-1) skew-symmetric matrix is always zero.

Application to 3x3 Matrix

Given a 3x3 matrix, which is both skew-symmetric and orthogonal, its determinant must be (-1). However, from the lemma, we know that the determinant of a 3x3 skew-symmetric matrix is zero. Hence, this matrix cannot exist.

It's worth noting that 2x2 and 4x4 matrices can indeed be both skew-symmetric and orthogonal. For example, a 2x2 skew-symmetric matrix can be:

[ A begin{pmatrix} 0 a -a 0 end{pmatrix} ]

And a 4x4 example could be:

[ A begin{pmatrix} 0 a b c -a 0 d e -b -d 0 f -c -e -f 0 end{pmatrix} ]

However, for 3x3 matrices, the combination of being both skew-symmetric and orthogonal is impossible due to the contradiction in the determinant value.

Conclusion

In conclusion, a 3x3 matrix that is both skew-symmetric and orthogonal does not exist. This is due to the contradiction between the determinant value required for both properties and the fact that the determinant of a 3x3 skew-symmetric matrix is zero. Understanding these concepts is vital for anyone working in linear algebra and matrix theory.