Toeplitz decomposition
Undergraduate (real matrices)
Consider a square real matrix $A$. The Toeplitz decomposition of $A$ is a sum
\[A = H + K\]where $H$ is a symmetric matrix and a $K$ is skew-symmetric.
Examples
Worked examples
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The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\)
: \(H = \begin{bmatrix} 2 & 5/2 \\ 5/2 & 3 \end{bmatrix}\), \(K = \begin{bmatrix} 0 & -3/2 \\ 3/2 & 0 \end{bmatrix}\)
Undergraduate
The Toeplitz decomposition of \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) splits $A$ into its symmetric part $H = \frac{1}{2}(A + A^{\top})$ and its skew-symmetric part $K = \frac{1}{2}(A - A^{\top})$, which average and split the off-diagonal entries $1$ and $4$.
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The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\)
: $H = A$, $K = 0$
Undergraduate
The matrix \(A = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) is symmetric positive definite, so it has every decomposition in the table, including a Cholesky decomposition. Its eigendecomposition, Schur decomposition, Jordan normal form, and singular value decomposition all coincide, and its polar and Toeplitz decompositions are trivial.
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The matrix \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
: \(H = \begin{bmatrix} 1 & 1/2 \\ 1/2 & 1 \end{bmatrix}\), \(K = \begin{bmatrix} 0 & 1/2 \\ -1/2 & 0 \end{bmatrix}\)
Undergraduate
The Toeplitz decomposition of \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) has the symmetric part \(H = \begin{bmatrix} 1 & 1/2 \\ 1/2 & 1 \end{bmatrix}\) and the skew-symmetric part \(K = \begin{bmatrix} 0 & 1/2 \\ -1/2 & 0 \end{bmatrix}\). $H$ and $K$ do not commute, because $A$ is not normal.
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The matrix \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
: $H = A$, $K = 0$
Undergraduate
The Toeplitz decomposition of \(A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) has $H = A$ and $K = 0$, because $A$ is symmetric.
Properties
- The Toeplitz decomposition exists and is unique.
- The Toeplitz decomposition of a square matrix $A$ is \[A = \frac{1}{2}(A + A^{\top}) + \frac{1}{2}(A - A^{\top})\]
- The matrix $A$ is normal ($A A^\top = A^\top A$) exactly when $H$ and $K$ commute.
Notes
Do not confuse the Toeplitz decomposition (named after Otto Toeplitz) with decompositions of Toeplitz matrices, such as the Vandermonde decomposition of a positive semidefinite Toeplitz matrix.
Distinctions involving this term
- Types of matrix decompositions: A matrix decomposition writes a matrix in terms of simpler matrices. Most are products: LU, PLU, and Cholesky decompositions use triangular factors, QR and singular value decompositions use orthogonal factors, and eigendecompositions, Schur decompositions, and Jordan normal forms are similarities $A = P M P^{-1}$ to a simpler matrix $M$ (diagonal, triangular, or made of Jordan blocks). In contrast, the Jordan–Chevalley decomposition is a sum of commuting diagonalizable and nilpotent parts, and the Toeplitz decomposition is a sum $H + K$ of a symmetric matrix $H$ and a skew-symmetric matrix $K$.