Enter the nine dimensionless entries of a real 3×3 matrix A, row by row. For example, a12 means row 1, column 2. The divisor is constrained to 2 and is not a matrix entry. Read the nine S outputs and nine K outputs in row order to assemble the two resulting matrices.
Transposing A swaps its row and column indices. Each symmetric entry is Sij = (aij + aji)/2, and each skew-symmetric entry is Kij = (aij − aji)/2. Thus Sᵀ = S and Kᵀ = −K. The diagonal of S equals the diagonal of A; the diagonal of K is zero.
Adding the formulas cancels the transpose terms, giving A = S + K. This decomposition is unique: a matrix that is both symmetric and skew-symmetric must be zero. Skew-symmetric is also called antisymmetric. See Wichita State’s matrix algebra notes, section 2.2.8.
| Row | A | S | K |
|---|---|---|---|
| 1 | 1, 2, 3 | 1, 3, 5 | 0, −1, −2 |
| 2 | 4, 5, 6 | 3, 5, 7 | 1, 0, −1 |
| 3 | 7, 8, 9 | 5, 7, 9 | 2, 1, 0 |
For the example, S12 + K12 = 3 − 1 = 2 and S21 + K21 = 3 + 1 = 4, recovering the corresponding entries of A. This is an additive decomposition into two matrices. LU, QR, Cholesky, SVD and diagonalization instead express matrices using products.
This tool accepts real 3×3 matrices, including singular matrices and the zero matrix. It reports entries rather than automatically classifying the input or testing identities with a numerical tolerance. Results are displayed to eight decimal places; rounding can obscure small nonzero entries or exact reconstruction. Extremely large intermediate values may produce unavailable results, which must not be interpreted as zero.