Symmetric Matrix — Properties & Examples

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Symmetric Matrix — Properties & Examples

TL;DR

A symmetric matrix is a square matrix that equals its own transpose, AT=AA^T = AAT=A, so the entry in row iii, column jjj always matches the entry in row jjj, column iii. This article covers the definition, the key properties (real eigenvalues, orthogonal eigenvectors, symmetric inverse), the symmetric-versus-skew-symmetric distinction, the decomposition theorem, and six worked examples.

Bhanzu Team Last updated on June 10, 2026 9 min read

What Is a Symmetric Matrix?

A symmetric matrix is a square matrix AAA that is equal to its own transpose:

AT=A.A^T = A.AT=A.

The transpose ATA^TAT is the matrix you get by swapping rows and columns — reflecting every entry across the main diagonal. So saying AT=AA^T = AAT=A is saying that this reflection changes nothing: the entry aija_{ij} equals the entry ajia{ji} for every pair of positions.

In entry form, a matrix is symmetric when aij=ajifor all i,j.a_{ij} = a{ji} \ \text{for all } i, j.aij=ajifor all i,j.

A symmetric matrix must be square, and the main-diagonal entries can be anything — they sit on the mirror line, so they map to themselves. Here is a symmetric 3\times 33 matrix; notice each off-diagonal entry appears twice, once on each side of the diagonal:

A=[172745259].A = \begin{bmatrix} 1 & 7 & 2
7 & 4 & 5
2 & 5 & 9 \end{bmatrix}.A=[172745259].

What Are the Properties of a Symmetric Matrix?

Symmetric matrices carry a set of properties that make them the best-behaved matrices to work with:

Real eigenvalues. Every eigenvalue of a real symmetric matrix is a real number — never complex. This is the spectral theorem's headline.
Orthogonal eigenvectors. Eigenvectors for distinct eigenvalues are mutually perpendicular, which connects directly to orthogonal matrices.
Orthogonally diagonalizable. A matrix is symmetric if and only if it can be written A=QΛQTA = Q\Lambda Q^TA=QΛQT, where QQQ is orthogonal and Λ\LambdaΛ is diagonal — the spectral theorem.
Sum stays symmetric. The sum (or difference) of two symmetric matrices is symmetric.
The inverse is symmetric. If a symmetric matrix is invertible, its inverse is symmetric too.
Powers stay symmetric. AnA^nAn is symmetric whenever AAA is.
Product caveat. The product ABABAB of two symmetric matrices is symmetric only if AAA and BBB commute (AB=BAAB = BAAB=BA) — symmetry is not automatically preserved under multiplication.

The real-eigenvalues property is the one that makes symmetric matrices central to physics and statistics — it guarantees the quantities they model behave sensibly.

What Is the Difference Between Symmetric and Skew-Symmetric Matrices?

A skew-symmetric (or antisymmetric) matrix is the mirror-image idea with a sign flip: it equals the negative of its transpose, AT=−AA^T = -AAT=−A. The two sit side by side:

Feature	Symmetric matrix	Skew-symmetric matrix
Transpose rule	AT=AA^T = AAT=A	AT=−AA^T = -AAT=−A
Entry condition	aij=ajia_{ij} = a{ji}aij=aji	aij=−ajia_{ij} = -a{ji}aij=−aji
Main diagonal	any values	all zeros (since aii=−aiia{ii} = -a{ii}aii=−aii)
Eigenvalues	all real	zero or purely imaginary

The forced-zero diagonal of a skew-symmetric matrix is the quickest way to tell the two apart: a skew-symmetric matrix must have zeros all down the diagonal, while a symmetric one can hold any diagonal values.

What Is the Symmetric and Skew-Symmetric Decomposition?

Here is the result that ties the two together: every square matrix splits uniquely into a symmetric part and a skew-symmetric part. For any square matrix AAA,

$$A = \underbrace{\tfrac{1}{2}\big(A + A^T\big)} {\text{symmetric}} + \underbrace{\tfrac{1}{2}\big(A - A^T\big)}{\text{skew-symmetric}}.$$

The first piece satisfies (12(A+AT))T=12(AT+A)\big(\tfrac{1}{2}(A + A^T)\big)^T = \tfrac{1}{2}(A^T + A)(21(A+AT))T=21(AT+A) — unchanged, so symmetric. The second flips sign under transpose, so skew-symmetric. Add them and the transposes cancel back to AAA. This decomposition (sometimes called the Toeplitz decomposition) is why symmetric matrices show up even when the original data isn't symmetric — you can always extract the symmetric part.

Examples of Symmetric Matrix

The set runs from a quick verification, through the most common symmetry-test mistake, to finding an unknown entry, building a symmetric matrix from any matrix, checking skew-symmetry, and the full decomposition.

Example 1

Verify that A=[3−1−15]A = \begin{bmatrix} 3 & -1
-1 & 5 \end{bmatrix}A=[3−1−15] is symmetric.

Transpose by swapping rows and columns: AT=[3−1−15]A^T = \begin{bmatrix} 3 & -1
-1 & 5 \end{bmatrix}AT=[3−1−15], which is identical to AAA. Equivalently, a12=−1=a21a_{12} = -1 = a_{21}a12=−1=a21.

Final answer: AT=AA^T = AAT=A, so AAA is symmetric. The single off-diagonal value matches its mirror.

Example 2

Is A=[241463158]A = \begin{bmatrix} 2 & 4 & 1
4 & 6 & 3
1 & 5 & 8 \end{bmatrix}A=[241463158] symmetric?

Wrong attempt. A student glances at the top-left block — a12=4=a21a_{12} = 4 = a_{21}a12=4=a21 — sees the first mirror pair match, and concludes "symmetric" without checking the rest.

Check every mirror pair, not just the first. Look at a23=3a{23} = 3a23=3 against a32=5a{32} = 5. Those do not match, so the symmetry fails — a single mismatched pair is enough to break it.

Correct. Since a23=3≠5=a32a{23} = 3 \neq 5 = a{32}a23=3=5=a32, the matrix is not symmetric.

Final answer: not symmetric. Symmetry requires all mirror pairs to match — one disagreement anywhere is fatal.

Example 3

Find xxx so that A=[5x2917274]A = \begin{bmatrix} 5 & x & 2
9 & 1 & 7
2 & 7 & 4 \end{bmatrix}A=[5x2917274] is symmetric.

For symmetry, a12=a21a_{12} = a_{21}a12=a21, which means x=9x = 9x=9. Check the other pairs: a13=2=a31a_{13} = 2 = a_{31}a13=2=a31 and a23=7=a32a_{23} = 7 = a_{32}a23=7=a32 already hold.

Final answer: x=9x = 9x=9. With that value, every mirror pair matches and the matrix is symmetric.

Example 4

Show that ATAA^T AATA is symmetric for A=[1203]A = \begin{bmatrix} 1 & 2
0 & 3 \end{bmatrix}A=[1203].

Compute ATAA^T AATA:

ATA=[1023][1203]=[12213].A^T A = \begin{bmatrix} 1 & 0
2 & 3 \end{bmatrix}\begin{bmatrix} 1 & 2
0 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2
2 & 13 \end{bmatrix}.ATA=[1023][1203]=[12213].

The result has a12=2=a21a_{12} = 2 = a_{21}a12=2=a21 — symmetric.

Final answer: ATA=[12213]A^T A = \begin{bmatrix} 1 & 2
2 & 13 \end{bmatrix}ATA=[12213], which is symmetric. In fact ATAA^T AATA is symmetric for any matrix AAA, because (ATA)T=AT(AT)T=ATA(A^T A)^T = A^T (A^T)^T = A^T A(ATA)T=AT(AT)T=ATA — a fact behind least-squares regression.

Example 5

Is B=[03−2−3042−40]B = \begin{bmatrix} 0 & 3 & -2
-3 & 0 & 4
2 & -4 & 0 \end{bmatrix}B=[03−2−3042−40] symmetric, skew-symmetric, or neither?

The diagonal is all zeros, a hint toward skew-symmetric. Check: b12=3b{12} = 3b12=3 and b21=−3b{21} = -3b21=−3, so b12=−b21b{12} = -b{21}b12=−b21. Every off-diagonal pair flips sign.

Final answer: BBB is skew-symmetric (BT=−BB^T = -BBT=−B). The all-zero diagonal is the giveaway — a symmetric matrix could have any diagonal, but a skew-symmetric one is forced to zero there.

Example 6

Decompose A=[2514]A = \begin{bmatrix} 2 & 5
1 & 4 \end{bmatrix}A=[2514] into symmetric and skew-symmetric parts.

First AT=[2154]A^T = \begin{bmatrix} 2 & 1
5 & 4 \end{bmatrix}AT=[2154]. The symmetric part is 12(A+AT)\tfrac{1}{2}(A + A^T)21(A+AT) and the skew-symmetric part is 12(A−AT)\tfrac{1}{2}(A - A^T)21(A−AT):

12(A+AT)=[2334],12(A−AT)=[02−20].\tfrac{1}{2}(A + A^T) = \begin{bmatrix} 2 & 3
3 & 4 \end{bmatrix}, \qquad \tfrac{1}{2}(A - A^T) = \begin{bmatrix} 0 & 2
-2 & 0 \end{bmatrix}.21(A+AT)=[2334],21(A−AT)=[02−20].

Final answer: A=[2334]+[02−20]A = \begin{bmatrix} 2 & 3
3 & 4 \end{bmatrix} + \begin{bmatrix} 0 & 2
-2 & 0 \end{bmatrix}A=[2334]+[02−20]. Add the two and the off-diagonal 3±23 ± 23±2 recovers the original 555 and 111 — the decomposition checks out.

Why the Symmetric Matrix Earns Its Place

"Why do the matrices that describe real physical quantities almost always come out symmetric?"

The deep result here is the spectral theorem — that every real symmetric matrix has real eigenvalues and an orthonormal set of eigenvectors — credited to Augustin-Louis Cauchy (1789–1857, France), who proved the realness of the eigenvalues in the 1820s while studying the axes of quadric surfaces. That single guarantee is why symmetric matrices anchor so much applied mathematics.

Where symmetric matrices do real work:

Statistics and machine learning. A covariance matrix is symmetric, because the covariance of XXX with YYY equals the covariance of YYY with XXX. Principal component analysis — the workhorse of dimensionality reduction — is the spectral decomposition of a symmetric covariance matrix.
Physics and engineering. Stress, strain, and inertia tensors are symmetric matrices; their real eigenvalues are the principal stresses or moments of inertia that engineers actually design around.
Quadratic forms and optimisation. The matrix of second derivatives (the Hessian) is symmetric, and its eigenvalues tell you whether a critical point is a minimum, maximum, or saddle.
Graph theory and networks. The adjacency matrix of an undirected graph is symmetric, because an edge between two nodes runs both ways.

Where Students Trip Up on the Symmetric Matrix

Mistake 1: Checking only one mirror pair

Where it slips in: Verifying symmetry on a 3\times 33 or larger matrix.

Don't do this: Confirm one off-diagonal pair matches and call the whole matrix symmetric.

The correct way: Check every mirror pair — a12a{12}a12 vs a21a{21}a21, a13a{13}a13 vs a31a{31}a31, a23a{23}a23 vs a32a{32}a32, and so on. A single mismatched pair breaks symmetry.

Mistake 2: Confusing symmetric with skew-symmetric

Where it slips in: Classifying a matrix with an all-zero diagonal.

Don't do this: Assume an all-zero diagonal means symmetric.

The correct way: An all-zero diagonal points toward skew-symmetric (AT=−AA^T = -AAT=−A), not symmetric. A symmetric matrix can have any diagonal; the forced-zero diagonal belongs to the skew case. The second-guesser who knows both definitions still mixes the sign — anchor on "skew flips the sign, and its diagonal must vanish."

Mistake 3: Assuming the product of symmetric matrices is symmetric

Where it slips in: Multiplying two symmetric matrices and expecting a symmetric result.

Don't do this: Treat ABABAB as symmetric just because AAA and BBB are.

The correct way: ABABAB is symmetric only if AAA and BBB commute (AB=BAAB = BAAB=BA). In general (AB)T=BTAT=BA≠AB(AB)^T = B^T A^T = BA \neq AB(AB)T=BTAT=BA=AB. The memorizer who learned "symmetric stays symmetric" overgeneralises from sums (which do stay symmetric) to products (which need not).

Key Takeaways

A symmetric matrix equals its own transpose, AT=AA^T = AAT=A, so aij=ajia_{ij} = a{ji}aij=aji — its entries mirror across the diagonal.
Real symmetric matrices have real eigenvalues and orthogonal eigenvectors, and are orthogonally diagonalizable (A=QΛQTA = Q\Lambda Q^TA=QΛQT).
A skew-symmetric matrix satisfies AT=−AA^T = -AAT=−A and must have an all-zero diagonal.
Every square matrix splits uniquely into a symmetric part 12(A+AT)\tfrac{1}{2}(A + A^T)21(A+AT) and a skew-symmetric part 12(A−AT)\tfrac{1}{2}(A - A^T)21(A−AT).
The product of two symmetric matrices is symmetric only when they commute.
Symmetric matrices model covariance, stress tensors, Hessians, and undirected graphs — the reason PCA and finite element analysis depend on them.

Practice These Before Moving On

Is [1662]\begin{bmatrix} 1 & 6
6 & 2 \end{bmatrix}[1662] symmetric? Justify with the transpose.
Find yyy so that [04y0]\begin{bmatrix} 0 & 4
y & 0 \end{bmatrix}[04y0] is skew-symmetric.
Decompose [3712]\begin{bmatrix} 3 & 7
1 & 2 \end{bmatrix}[3712] into symmetric and skew-symmetric parts.

Answer to Question 1: AT=AA^T = AAT=A (the 666s mirror), so yes, symmetric. Answer to Question 2: y=−4y = -4y=−4 (skew needs a21=−a12a{21} = -a{12}a21=−a12). Answer to Question 3: symmetric part [3442]\begin{bmatrix} 3 & 4
4 & 2 \end{bmatrix}[3442], skew-symmetric part [03−30]\begin{bmatrix} 0 & 3
-3 & 0 \end{bmatrix}[03−30]. If Question 2 tempted you toward y=4y = 4y=4, return to Mistake 2.