Concept 02 / 09Matrices and Determinants
Transpose, Symmetric and Skew-Symmetric Matrices
The transpose and the reversal rule (AB)^T = B^T A^T, symmetric and skew-symmetric matrices, why the diagonal of a skew-symmetric matrix is zero and its determinant of odd order vanishes, and the unique split of any square matrix into symmetric and skew-symmetric parts.
The transpose (also written ) of an matrix is the matrix obtained by turning rows into columns: . Properties: , , and, most importantly,
.
The order reverses. To see why, the entry of is the entry of , which is row of dotted with column of , which is row of dotted with column of , the entry of . The rule extends to , and it is also consistent with sizes: if is and is , then is , which is defined, while usually is not.
Symmetric and skew-symmetric. A square matrix is symmetric if , that is, : it is a mirror image of itself across the diagonal. It is skew-symmetric if , that is, . Setting in the skew condition gives , so every diagonal entry of a skew-symmetric matrix is . The only matrix that is both symmetric and skew-symmetric is .
Closure facts, each a one-line check with the transpose rules:
is symmetric and is skew-symmetric.
and are always symmetric (for any, even non-square, ).
If are symmetric, is symmetric if and only if , since .
If is symmetric, so is for any of suitable order; if is skew-symmetric, so is .
Powers: if is symmetric, all powers are symmetric; if is skew-symmetric, odd powers are skew-symmetric and even powers are symmetric.
Decomposition. Every square matrix can be written uniquely as a symmetric matrix plus a skew-symmetric matrix:
.
Existence is the formula itself. Uniqueness: if with symmetric and skew, then is both symmetric and skew-symmetric, hence . This is the matrix version of splitting a function into its even and odd parts.
Determinant of a skew-symmetric matrix. Since and for an matrix, a skew-symmetric satisfies . If is odd, , so . A skew-symmetric matrix of odd order is therefore always singular. For even order the determinant is a perfect square of a polynomial in the entries; for example, .
Counting. For matrices with entries from a set of values, a symmetric matrix is determined by the entries on and above the diagonal, giving matrices. A skew-symmetric matrix is determined by the entries above the diagonal (the diagonal is forced to be ), provided the entry set is closed under negation.
Key Takeaways:
- ; ; .
- Symmetric: . Skew-symmetric: , with all diagonal entries .
- symmetric; skew; and symmetric.
- Product of symmetric matrices is symmetric iff they commute.
- , uniquely.
- A skew-symmetric matrix of odd order has determinant .
- Symmetric matrices have free entries; skew-symmetric have .
- Pitfall: writing .
Brain-Teaser 1: and are symmetric matrices of the same order. Classify and .
Solution: , so is skew-symmetric. Similarly , so is symmetric. As a consequence, the diagonal entries of are all , and for odd order . Applying the transpose rule and comparing with the original is the universal method for classification questions.
Brain-Teaser 2: Express as the sum of a symmetric and a skew-symmetric matrix.
Solution: . The symmetric part is . The skew part is . Check: . A quick shortcut: the off-diagonal entries of are the averages of the mirror pairs (), and those of are the half-differences (), while the diagonal goes entirely to .
Brain-Teaser 3: If is skew-symmetric and is any column vector of the right size, show that .
Solution: is a matrix, so it equals its own transpose: . So . Written out, , and the terms and cancel in pairs while the diagonal terms are . The trick of transposing a scalar is short but powerful, and it is used again for quadratic forms and orthogonal matrices.
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