Symmetric Matrices And Their Entries

What Makes a Matrix Symmetric?

Some square matrices have a built-in reflection: an entry above the main diagonal matches the entry the same distance below it. This pattern is called symmetry. It is compactly expressed using the transpose, but it can also be checked one entry at a time. Both descriptions are useful: the transpose equation gives a concise test, while the entry condition makes it straightforward to recognize and construct examples.

In this tutorial, matrices have real entries. If \(\mathbf{A}\) is an \(m\times n\) matrix, its transpose \(\mathbf{A}^{\top}\) is the \(n\times m\) matrix whose entry in row \(i\), column \(j\) is the entry \(a_{ji}\) of \(\mathbf{A}\). Thus, transposition interchanges the row and column positions of each entry.

The square requirement is part of the definition. An \(m\times n\) matrix has transpose of size \(n\times m\), so an equality between the matrix and its transpose is possible only if the two sizes agree. For a square matrix, symmetry can be checked by comparing each entry with its reflected partner: \(a_{ij}\) must equal \(a_{ji}\). Entries on the main diagonal are their own partners and impose no comparison with a different entry.

Proof. By the definition of transpose, the entry in row \(i\), column \(j\) of \(\mathbf{A}^{\top}\) is \(a_{ji}\). Two matrices of the same size are equal exactly when their corresponding entries are equal. Therefore, \(\mathbf{A}^{\top}=\mathbf{A}\) holds exactly when \(a_{ji}=a_{ij}\) for every \(i,j\). This proves both directions. \(\square\)

For \(i=j\), the condition says \(a_{ii}=a_{ii}\), which is always true. The meaningful checks are the pairs with \(i\ne j\). One need not compare every entry separately: checking each pair on opposite sides of the diagonal is sufficient.

Testing Symmetry in Small Matrices

A single unequal pair is enough to rule out symmetry. When testing a matrix, it is efficient to stop as soon as such a pair appears. To prove that a matrix is symmetric, however, all off-diagonal pairs must match, or one must verify the full transpose equation.

Constructing Symmetric Matrices

The entry test gives a direct construction method. Choose the entries on the main diagonal and on one side of it; then copy each chosen off-diagonal entry into the reflected position on the other side. Either the upper or the lower triangle may be chosen first. The resulting matrix is symmetric because each reflected pair has equal entries.

Proof. For each pair \(i<j\), the entry test requires \(a_{ji}=a_{ij}\). Thus, once the entry above the diagonal, \(a_{ij}\), is chosen, the corresponding entry below the diagonal is fixed. Each diagonal entry \(a_{ii}\) may be chosen independently because its symmetry condition compares it only with itself. There are \(n\) diagonal positions. Above the diagonal, row \(i\) has \(n-i\) positions, so the number of positions strictly above it is \((n-1)+(n-2)+\cdots+1=n(n-1)/2\). The total number of determining entries is therefore \(n+n(n-1)/2=n(n+1)/2\). Conversely, assigning those entries and copying each one to its reflected position makes every pair equal, so it produces a symmetric matrix. \(\square\)

The theorem also explains the number of choices in this example: a symmetric \(3\times3\) matrix has \(3(3+1)/2=6\) determining entries. Here these are three diagonal entries and three entries above the diagonal. The other three entries are not additional choices; symmetry forces them to match.

How Symmetry Behaves Under Addition and Scaling

Symmetric matrices of the same size remain symmetric when added or multiplied by a real scalar. This follows directly from the entry condition, without needing any further matrix properties.

Proof. Since \(\mathbf{A}\) and \(\mathbf{B}\) are symmetric, \(a_{ij}=a_{ji}\) and \(b_{ij}=b_{ji}\) for all \(i,j\). The \((i,j)\)-entry of \(\mathbf{A}+\mathbf{B}\) is \(a_{ij}+b_{ij}\), while its \((j,i)\)-entry is \(a_{ji}+b_{ji}\). These are equal because \(a_{ij}=a_{ji}\) and \(b_{ij}=b_{ji}\). Thus \(\mathbf{A}+\mathbf{B}\) is symmetric. The \((i,j)\)-entry of \(c\mathbf{A}\) is \(ca_{ij}\), and the \((j,i)\)-entry is \(ca_{ji}\). They are equal because \(a_{ij}=a_{ji}\), so \(c\mathbf{A}\) is symmetric. This includes \(c=0\), when \(c\mathbf{A}\) is the zero matrix, whose entries all match their reflected entries. \(\square\)

The requirement that the two matrices have the same size matters for addition: the sum is defined only when their dimensions agree. The result concerns addition and scalar multiplication; symmetry should not be assumed automatically for other matrix operations.

Common Pitfall: Comparing the Wrong Entries

A frequent error is to compare entries that happen to be in the same row rather than entries reflected across the main diagonal. For example, consider \(\mathbf{E}=\begin{bmatrix}2&5\\1&3\end{bmatrix}\). A wrong check might compare \(e_{11}=2\) and \(e_{22}=3\), notice that they differ, and conclude that \(\mathbf{E}\) is not symmetric for that reason. The conclusion happens to be correct, but the calculation is not a valid test: symmetry does not require the two diagonal entries to match.

The relevant off-diagonal comparison is \(e_{12}=5\) against \(e_{21}=1\). Since \(5\ne1\), the matrix is nonsymmetric. Directly,

The correction is to compare positions \((i,j)\) and \((j,i)\), not unrelated positions. In a \(2\times2\) matrix, this means checking the two off-diagonal entries; in a larger matrix, check every reflected pair. Diagonal entries may take any real values without violating symmetry.

Symmetry is therefore a precise entry pattern, not a requirement that a matrix look visually balanced in some other way. The transpose equation provides the definition, the entry test gives a practical method, and the construction rule shows exactly which entries can be chosen freely.