Symmetric matrix
High school
A square matrix $A$ is called symmetric if $A$ is unchanged when flipped across the main diagonal: for every $i$ and $j$, the entry in row $i$ and column $j$ of $A$ equals the entry in row $j$ and column $i$ of $A$.
Examples
Examples
Non-examples
Things that may be mistaken for examples.
- The matrix \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) High school
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The matrix \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\)
Undergraduate
The matrix \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\) is not symmetric, because $A_{12} = 1 \neq 4 = A_{21}$.
Properties
- A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors (the spectral theorem), so $A = Q \Lambda Q^{\top}$ with $Q$ orthogonal and $\Lambda$ diagonal.
- Every square matrix is the sum of a symmetric matrix and a skew-symmetric matrix, called its Toeplitz decomposition: \[A = \frac{1}{2}(A + A^{\top}) + \frac{1}{2}(A - A^{\top}).\]