what is symmetric matrix

A symmetric matrix is a square matrix that is equal to its transpose: A=ATA=A^{T}A=AT.

Quick Scoop: What is a symmetric matrix?

• A symmetric matrix is always square (same number of rows and columns).

• Its entries mirror across the main diagonal, so aij=ajia_{ij}=a_{ji}aij​=aji​ for all i,ji,ji,j.

• Example:

A=[2−14−130405]A=\begin{bmatrix} 2&-1&4\\ -1&3&0\\ 4&0&5 \end{bmatrix}A=​2−14​−130​405​​

Here, the entry above the diagonal matches the one below it in each position, so AAA is symmetric.

Key properties (why we care)

• The sum and difference of two symmetric matrices are symmetric.

• Their product is symmetric only if they commute (i.e. AB=BAAB=BAAB=BA).

• Powers of a symmetric matrix, like A2,A3A^{2},A^{3}A2,A3, are also symmetric.

• All eigenvalues of a real symmetric matrix are real, and eigenvectors for different eigenvalues are orthogonal.

• Real symmetric matrices can always be diagonalized (spectral theorem), which is crucial in applications like PCA and optimization.

Quick contrast: symmetric vs skew-symmetric

• Symmetric: AT=AA^{T}=AAT=A.

• Skew-symmetric: AT=−AA^{T}=-AAT=−A and all diagonal entries are 0.

Any square matrix can be decomposed as:

B=12(B+BT)+12(B−BT),B=\tfrac{1}{2}(B+B^{T})+\tfrac{1}{2}(B-B^{T}),B=21​(B+BT)+21​(B−BT),

where the first term is symmetric and the second is skew-symmetric.

Mini SEO bits

• Focus phrase “what is symmetric matrix” : a symmetric matrix is a square matrix equal to its transpose, with mirrored entries across the main diagonal.

• Common use cases: machine learning covariance matrices, graph Laplacians, quadratic forms in optimization, and many numerical algorithms.