Is the Product of Two Symmetric Matrices Similar to a Symmetric Matrix?

The statement is false for arbitrary real symmetric matrices. Symmetry of the factors does not guarantee that their product has the spectral properties of a symmetric matrix. A short counterexample settles the claim; a positive-definite factor gives a useful condition under which the conclusion is true.

A counterexample

Take the real symmetric matrices

$$A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \qquad B=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

Their product is

$$AB=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$

Its characteristic polynomial is

$$\det(\lambda I-AB)=\det\begin{pmatrix}\lambda&-1\\1&\lambda\end{pmatrix}=\lambda^2+1,$$

so its eigenvalues are $i$ and $-i$. But every real symmetric matrix has real eigenvalues, by the spectral theorem. Similar matrices have the same characteristic polynomial and therefore the same eigenvalues. Thus $AB$ cannot be similar over the reals—or even over the complex numbers—to a real symmetric matrix.

This example also shows why it is not enough to note that each factor is diagonalizable: the product can have nonreal eigenvalues even though both factors are real symmetric.

A condition that makes the claim true

Suppose $A$ is real symmetric and positive definite, and $B$ is real symmetric of the same size. Positive definiteness means $x^T A x>0$ for every nonzero real vector $x$. The spectral theorem gives a symmetric positive-definite square root $A^{1/2}$, which is invertible. Then

$$\begin{aligned} A^{-1/2}(AB)A^{1/2} &=A^{1/2}BA^{1/2}. \end{aligned}$$

The right-hand side is symmetric, since

$$(A^{1/2}BA^{1/2})^T=A^{1/2}B^TA^{1/2}=A^{1/2}BA^{1/2}.$$

Therefore $AB$ is similar to the symmetric matrix $A^{1/2}BA^{1/2}$. The same argument works if $B$ is positive definite: conjugate $AB$ using $B^{1/2}$ to obtain $B^{1/2}AB^{1/2}$.

Worked example

Let

$$A=\begin{pmatrix}4&0\\0&1\end{pmatrix}, \qquad B=\begin{pmatrix}2&1\\1&3\end{pmatrix}.$$

Both are symmetric, and $A$ is positive definite. Since $A^{1/2}=\begin{pmatrix}2&0\\0&1\end{pmatrix}$,

$$AB=\begin{pmatrix}8&4\\1&3\end{pmatrix}, \qquad A^{1/2}BA^{1/2}=\begin{pmatrix}8&2\\2&3\end{pmatrix}.$$

The second matrix is symmetric, and the calculation above gives

$$A^{-1/2}(AB)A^{1/2}=\begin{pmatrix}8&2\\2&3\end{pmatrix}.$$

So the matrices are similar. As a check, both have characteristic polynomial $\lambda^2-11\lambda+20$ and eigenvalues $(11\pm\sqrt{41})/2$.

The exact criterion

For any real square matrix $M$, the following are equivalent:

• $M$ is similar over the reals to a real symmetric matrix.
• $M$ is diagonalizable over the reals and all its eigenvalues are real.

One direction follows from the spectral theorem: a real symmetric matrix is orthogonally diagonalizable, and similarity preserves diagonalizability and eigenvalues. For the other direction, if $M=PDP^{-1}$ for an invertible real matrix $P$ and a real diagonal matrix $D$, then $D$ itself is symmetric. This criterion applies to $M=AB$; symmetry of $A$ and $B$ alone does not ensure it.

Common mistakes

• Assuming the product is symmetric. In general, $(AB)^T=BA$, not $AB$. The product is symmetric exactly when $AB=BA$.
• Confusing symmetry with similarity to a symmetric matrix. A matrix can be nonsymmetric yet similar to a symmetric one, as in the positive-definite example. Conversely, the counterexample’s product is nonsymmetric and cannot be similar to a real symmetric matrix because its eigenvalues are nonreal.
• Dropping the positive-definite assumption. The square-root conjugation uses $A^{-1/2}$, so it requires $A$ to be invertible. If a factor is only positive semidefinite, that argument does not apply directly; check the product’s diagonalizability and eigenvalues instead.
• Ignoring the field. This discussion concerns real symmetric matrices and similarity to a real symmetric matrix. Similarity preserves the characteristic polynomial regardless of whether the change-of-basis matrix is real or complex, so the nonreal eigenvalues in the counterexample rule out either kind of similarity to a real symmetric target.