Computing the Inverse of an Eigenvector Matrix

Why Compute the Inverse of an Eigenvector Matrix?

When a matrix is diagonalizable, its chosen eigenvectors can be placed in the columns of a matrix \(\mathbf{P}\). The order of those columns is important: it fixes the order of the basis vectors and must match the order of the corresponding eigenvalues. Computing \(\mathbf{P}^{-1}\) gives a practical way to change coordinates. The matrix \(\mathbf{P}\) turns coordinates in the eigenvector basis into standard coordinates; its inverse carries a vector back to its coordinates in that basis.

Throughout, vectors and matrices have real entries. An ordered basis \(\mathcal{B}=(\mathbf{v}_1,\dots,\mathbf{v}_n)\) of \(\mathbb{R}^n\) is a list that is linearly independent and spans \(\mathbb{R}^n\). If \(\mathbf{P}=\begin{bmatrix}\mathbf{v}_1&\cdots&\mathbf{v}_n\end{bmatrix}\), then multiplying \(\mathbf{P}\) by a column of coefficients forms the corresponding linear combination of the basis vectors. The key task is to find the matrix that reverses this operation.

In a diagonalization, the columns of \(\mathbf{P}\) are eigenvectors, and each diagonal entry of the diagonal matrix is the eigenvalue belonging to the eigenvector in the corresponding column. The earlier result that a matrix is diagonalizable exactly when it has an eigenbasis ensures that such a square basis matrix is invertible. This tutorial focuses on computing its inverse and interpreting the coordinate change, rather than testing or verifying a diagonalization.

Row Reduction for the Inverse

To compute \(\mathbf{P}^{-1}\), augment \(\mathbf{P}\) with the identity matrix and row-reduce the left side to the identity. Apply every row operation to the entire augmented matrix, including the right-hand side. The block that ends up on the right is \(\mathbf{P}^{-1}\). This method works when \(\mathbf{P}\) is invertible; here, that condition is met because its columns are a basis.

Proof. For each column \(\mathbf{e}_j\) of \(\mathbf{I}_n\), the basis property gives a unique coefficient column \(\mathbf{c}_j\) such that \(\mathbf{P}\mathbf{c}_j=\mathbf{e}_j\). Collect these columns into \(\mathbf{C}=\begin{bmatrix}\mathbf{c}_1&\cdots&\mathbf{c}_n\end{bmatrix}\). Then \(\mathbf{P}\mathbf{C}=\mathbf{I}_n\), column by column. Row operations on an augmented matrix perform the same operations on each right-hand-side column and preserve the solution sets of the corresponding systems. Reducing the left block to \(\mathbf{I}_n\) therefore turns each system \(\mathbf{P}\mathbf{c}_j=\mathbf{e}_j\) into \(\mathbf{I}_n\mathbf{c}_j=\mathbf{c}_j\). The right block after reduction has columns \(\mathbf{c}_1,\dots,\mathbf{c}_n\), so it is \(\mathbf{C}\). Since \(\mathbf{P}\mathbf{C}=\mathbf{I}_n\), \(\mathbf{C}\) is the inverse of \(\mathbf{P}\), as required. \(\square\)

There are two useful checks. First, multiply the proposed inverse on the left by \(\mathbf{P}\) and confirm the result is \(\mathbf{I}_n\). Second, for a specific vector \(\mathbf{x}\), calculate \(\mathbf{c}=\mathbf{P}^{-1}\mathbf{x}\), then check that \(\mathbf{P}\mathbf{c}=\mathbf{x}\). The first checks the matrix calculation; the second checks the coordinate interpretation in a particular case.

Worked Inverses and Coordinate Changes

How the Two Coordinate Directions Work

Proof. By the definition of the coordinate vector, \(\mathbf{x}=c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n\), where \([\mathbf{x}]_{\mathcal{B}}=\begin{bmatrix}c_1&\cdots&c_n\end{bmatrix}^{\top}\). By the definition of matrix multiplication, \(\mathbf{P}[\mathbf{x}]_{\mathcal{B}}=c_1\mathbf{v}_1+\cdots+c_n\mathbf{v}_n=\mathbf{x}\). Since \(\mathbf{P}^{-1}\mathbf{P}=\mathbf{I}_n\), multiplying the equality \(\mathbf{x}=\mathbf{P}[\mathbf{x}]_{\mathcal{B}}\) by \(\mathbf{P}^{-1}\) gives \(\mathbf{P}^{-1}\mathbf{x}=[\mathbf{x}]_{\mathcal{B}}\). \(\square\)

The direction matters. A coordinate column such as \(\begin{bmatrix}2\\0\\1\end{bmatrix}\) lists coefficients relative to the chosen basis; multiplying by \(\mathbf{P}\) builds the corresponding vector in standard coordinates. In contrast, \(\mathbf{P}^{-1}\mathbf{x}\) starts with a vector in standard coordinates and returns its coefficients relative to the ordered basis. These are inverse operations, not interchangeable ones.

For a diagonalization, the column order must be kept consistent throughout. If the first column of \(\mathbf{P}\) is an eigenvector for \(\lambda_1\) and the second is an eigenvector for \(\lambda_2\), then the first and second coordinates returned by \(\mathbf{P}^{-1}\) are coefficients on those same eigenvectors, in that order. Reordering columns changes the coordinate system and requires recomputing or correspondingly reordering the inverse.

Common Mistake: Using the Transpose

A common error is to assume that \(\mathbf{P}^{-1}=\mathbf{P}^{\top}\) for every eigenvector matrix. The transpose is the inverse when the columns are orthonormal, but eigenvectors in a basis need not be orthonormal. For the last example, a wrong calculation would use \(\mathbf{P}^{\top}=\begin{bmatrix}1&1\\2&1\end{bmatrix}\). It gives \(\mathbf{P}^{\top}\begin{bmatrix}3\\2\end{bmatrix}=\begin{bmatrix}5\\8\end{bmatrix}\), which is not the coordinate vector: multiplying back gives \(\mathbf{P}\begin{bmatrix}5\\8\end{bmatrix}=\begin{bmatrix}21\\13\end{bmatrix}\ne\begin{bmatrix}3\\2\end{bmatrix}\). The correction is to row-reduce \(\begin{bmatrix}\mathbf{P}&\mathbf{I}\end{bmatrix}\), giving \(\mathbf{P}^{-1}=\begin{bmatrix}-1&2\\1&-1\end{bmatrix}\) and coordinates \(\begin{bmatrix}1\\1\end{bmatrix}\), which pass the reconstruction check.

Before using an inverse in a diagonalization calculation, make these checks:

• Confirm that the columns of \(\mathbf{P}\) are in the intended eigenvector order.
• Apply every row operation to both blocks of the augmented matrix.
• Multiply \(\mathbf{P}\mathbf{P}^{-1}\) to check the inverse.
• For a coordinate calculation, reconstruct the vector by multiplying the coordinates by \(\mathbf{P}\).

These checks distinguish a correct inverse from a plausible-looking matrix and confirm that the entries have the intended coordinate meaning.