In one line
A square matrix is symmetric if \(A^T=A\) and skew-symmetric if \(A^T=-A\); every square matrix equals \(\frac12(A+A^T)+\frac12(A-A^T)\), a symmetric part plus a skew-symmetric part.
Explained simply
The transpose \(A^T\) flips a matrix across its main diagonal: rows become columns. A symmetric matrix looks the same in a mirror placed along the diagonal, like a distance table between cities where the distance from Mumbai to Pune equals Pune to Mumbai. A skew-symmetric matrix flips sign in the mirror, so its diagonal entries must be zero (a number equal to its own negative is 0).
Technically, \(A=[a_{ij}]\) is symmetric if \(a_{ij}=a_{ji}\) for all \(i,j\), and skew-symmetric if \(a_{ij}=-a_{ji}\), which forces \(a_{ii}=0\). These are pre-requisite definitions for this module and lead directly to their complex versions, Hermitian and skew-Hermitian matrices.
Symmetric matrices appear everywhere in engineering: stiffness matrices of structures, conductance matrices in circuit analysis and covariance matrices in statistics are all symmetric. A quick test is to look across the main diagonal and check that matching entries are equal (symmetric) or opposite in sign (skew-symmetric).
How it works
- Find \(A^T\) by writing rows as columns.
- Compute \(P=\frac12(A+A^T)\). Then \(P^T=\frac12(A^T+A)=P\), so \(P\) is symmetric.
- Compute \(Q=\frac12(A-A^T)\). Then \(Q^T=\frac12(A^T-A)=-Q\), so \(Q\) is skew-symmetric.
- Check \(P+Q=A\). This split is unique.
Key formula
\(A=\underbrace{\tfrac12(A+A^T)}_{\text{symmetric}}+\underbrace{\tfrac12(A-A^T)}_{\text{skew-symmetric}}\). Useful rules: \((A^T)^T=A\), \((A+B)^T=A^T+B^T\), \((AB)^T=B^TA^T\).
Quick example
\(A=\begin{bmatrix}1&2\\4&3\end{bmatrix}\), \(A^T=\begin{bmatrix}1&4\\2&3\end{bmatrix}\).
\(P=\frac12\begin{bmatrix}2&6\\6&6\end{bmatrix}=\begin{bmatrix}1&3\\3&3\end{bmatrix}\) (symmetric), \(\ Q=\frac12\begin{bmatrix}0&-2\\2&0\end{bmatrix}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\) (skew-symmetric).
Check: \(P+Q=\begin{bmatrix}1&2\\4&3\end{bmatrix}=A\).
Common mistakes
- Writing \((AB)^T=A^TB^T\); the order reverses: \(B^TA^T\).
- Thinking a skew-symmetric matrix can have non-zero diagonal entries.
- Forgetting the factor \(\frac12\) in \(P\) and \(Q\).
How MU asks it
As a pre-requisite it is not usually a full question, but a 3-mark part may ask "Express \(A\) as the sum of a symmetric and a skew-symmetric matrix". Show \(A^T\), both \(P\) and \(Q\), verify their properties in one line, and check \(P+Q=A\). The same pattern with conjugate transpose is a common Hermitian question.