Constructing Symmetric Matrices from Eigenvalue Data

From Eigenvalue Data to a Matrix

Eigenvalue problems are often presented in the forward direction: given a matrix, find its eigenvalues and eigenvectors. The reverse direction is also useful. If we choose real eigenvalues and an orthonormal basis of eigenvectors, we can construct a matrix that has exactly those eigenpairs. The construction also makes symmetry visible and provides a direct way to check how the matrix changes—or does not change—when the eigenvectors are represented differently.

Throughout, the matrices are real. A list \(\mathbf{q}_1,\dots,\mathbf{q}_n\) is an orthonormal basis of \(\mathbb{R}^n\) when \(\mathbf{q}_i^{\top}\mathbf{q}_j=0\) for \(i\ne j\), \(\mathbf{q}_i^{\top}\mathbf{q}_i=1\), and the list is a basis. Put these vectors in order as the columns of \(\mathbf{Q}\). Then \(\mathbf{Q}^{\top}\mathbf{Q}=\mathbf{I}_n\); a square matrix with this property is called orthogonal. Put the real numbers \(\lambda_1,\dots,\lambda_n\) in a diagonal matrix \(\mathbf{D}\), in the same order as their corresponding vectors.

The order of the data matters: the \(j\)th diagonal entry of \(\mathbf{D}\) is paired with the \(j\)th column of \(\mathbf{Q}\). A correct construction therefore begins by deciding which eigenvalue belongs to each chosen eigenvector. The following result explains why the formula works and when the full data determine the matrix uniquely.

Proof. Since \(\mathbf{Q}^{\top}\mathbf{Q}=\mathbf{I}_n\), the columns of \(\mathbf{Q}\) form a basis and \(\mathbf{Q}\mathbf{Q}^{\top}=\mathbf{I}_n\) as well. The matrix \(\mathbf{D}\) is diagonal with real entries, so \(\mathbf{D}^{\top}=\mathbf{D}\). Therefore \((\mathbf{Q}\mathbf{D}\mathbf{Q}^{\top})^{\top}=\mathbf{Q}\mathbf{D}^{\top}\mathbf{Q}^{\top}=\mathbf{Q}\mathbf{D}\mathbf{Q}^{\top}\), proving symmetry. For each \(j\), \(\mathbf{Q}^{\top}\mathbf{q}_j=\mathbf{e}_j\), where \(\mathbf{e}_j\) is the \(j\)th standard basis vector. Thus \(\mathbf{A}\mathbf{q}_j=\mathbf{Q}\mathbf{D}\mathbf{e}_j=\lambda_j\mathbf{Q}\mathbf{e}_j=\lambda_j\mathbf{q}_j\). To prove uniqueness, let \(\mathbf{B}\) be any matrix satisfying \(\mathbf{B}\mathbf{q}_j=\lambda_j\mathbf{q}_j\) for all \(j\). Collecting these equations as columns gives \(\mathbf{B}\mathbf{Q}=\mathbf{Q}\mathbf{D}\). Multiplying on the right by \(\mathbf{Q}^{\top}\) gives \(\mathbf{B}=\mathbf{Q}\mathbf{D}\mathbf{Q}^{\top}\). Hence no other matrix satisfies all the prescribed equations. \(\square\)

The formula can also be read as a sequence of operations: \(\mathbf{Q}^{\top}\) takes coordinates in the orthonormal eigenvector basis, \(\mathbf{D}\) multiplies each coordinate by its assigned eigenvalue, and \(\mathbf{Q}\) returns to standard coordinates. This interpretation helps keep the order and pairing straight.

Constructing and Checking the Matrix

Signs, Ordering, and Repeated Eigenvalues

Changing the sign of an eigenvector does not change the matrix, provided its eigenvalue stays paired with it. Reordering the data also leaves the matrix unchanged when each eigenvalue moves together with its eigenvector. These facts follow from a single invariance principle.

Proof. Since \(\mathbf{U}\) is orthogonal, \(\mathbf{U}\mathbf{U}^{\top}=\mathbf{I}_n\). The proposed construction is \(\mathbf{Q}\mathbf{U}\mathbf{D}\mathbf{U}^{\top}\mathbf{Q}^{\top}\). From \(\mathbf{U}\mathbf{D}=\mathbf{D}\mathbf{U}\), multiplying on the left by \(\mathbf{U}^{\top}\) gives \(\mathbf{D}=\mathbf{U}^{\top}\mathbf{D}\mathbf{U}\); multiplying this equality on the right by \(\mathbf{U}^{\top}\) gives \(\mathbf{D}\mathbf{U}^{\top}=\mathbf{U}^{\top}\mathbf{D}\). Consequently \(\mathbf{U}\mathbf{D}\mathbf{U}^{\top}=\mathbf{D}\mathbf{U}\mathbf{U}^{\top}=\mathbf{D}\), proving the claimed equality. For sign changes, take \(\mathbf{U}\) diagonal with entries \(1\) or \(-1\); it commutes with diagonal \(\mathbf{D}\). For a permutation, let \(\mathbf{P}\) be the matrix that permutes standard basis vectors, take \(\mathbf{Q}'=\mathbf{Q}\mathbf{P}\), and move the diagonal entries with them by setting \(\mathbf{D}'=\mathbf{P}^{\top}\mathbf{D}\mathbf{P}\). Then \(\mathbf{Q}'\mathbf{D}'\mathbf{Q}'^{\top}=\mathbf{Q}\mathbf{P}\mathbf{P}^{\top}\mathbf{D}\mathbf{P}\mathbf{P}^{\top}\mathbf{Q}^{\top}=\mathbf{Q}\mathbf{D}\mathbf{Q}^{\top}\). \(\square\)

If an eigenvalue is repeated, the choice of orthonormal basis within its eigenspace can change as well. A change of orthonormal basis within that eigenspace acts by an orthogonal transformation on the corresponding columns. Because the associated diagonal entries of \(\mathbf{D}\) are equal, that transformation commutes with \(\mathbf{D}\), so the theorem shows that the constructed matrix is unchanged.

A Pairing Mistake to Avoid

It is not enough to permute the columns of \(\mathbf{Q}\) while leaving the diagonal entries of \(\mathbf{D}\) fixed. That changes which eigenvalue is assigned to each direction. For the two-dimensional example above, swap the columns of \(\mathbf{Q}\) but keep \(\mathbf{D}=\operatorname{diag}(5,1)\). The resulting matrix is \(\begin{bmatrix}3&-2\\-2&3\end{bmatrix}\), not the original \(\begin{bmatrix}3&2\\2&3\end{bmatrix}\). Indeed, the new first column is the old \(\mathbf{q}_2\), so the new matrix multiplies that vector by \(5\) rather than \(1\).

The correction is to swap the diagonal entries at the same time: use \(\mathbf{D}'=\operatorname{diag}(1,5)\). Then the original vector–eigenvalue pairings are restored, and the formula returns \(\begin{bmatrix}3&2\\2&3\end{bmatrix}\). A useful final audit is to check that the columns of \(\mathbf{Q}\) are orthonormal, that \(\mathbf{D}\) lists the eigenvalues in matching order, and that multiplication confirms \(\mathbf{A}\mathbf{q}_j=\lambda_j\mathbf{q}_j\) for each column.

The Spectral Theorem states that every real symmetric matrix has an orthonormal basis of real eigenvectors. Its proof comes in the earlier material on the Spectral Theorem. The construction here gives the reverse direction directly: any chosen real eigenvalues paired with an orthonormal basis produce a real symmetric matrix. Thus the formula is both a practical construction and a precise description of how a full set of orthonormal eigenpair data determines a matrix.