Short Notes: Eigendecomposition of a Matrix

A vector \(\mathbf{v}_i \in \mathbb{C}^{n}\) is called the \(i\)-th eigenvector of a matrix \(\mathbf{A} \in \mathbb{R}^{n \times n}\), if it satisfies the simple equation

\[\begin{align} \mathbf{A} \mathbf{v}_i = \lambda_i \mathbf{v}_i, \label{eq:eigvalues} \end{align}\]

for a scalar value \(\lambda_i \in \mathbb{C}\), called an eigenvalue. (Assuming the matrix \(\mathbf{A}\) is real-valued, the eigenvalues and eigenvectors might still be complex.) Let us further assume that the \(n\) eigenvectors of matrix \(\mathbf{A}\) are linearly independent.

We can now ‘horizontally’ stack the eigenvectors into a matrix \(\mathbf{Q} \in \mathbb{C}^{n \times n}\):

\[\begin{align} \mathbf{Q} = \big[\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \big]. \label{eq:Q} \end{align}\]

Multiplying \(\mathbf{A}\) with \(\mathbf{Q}\) gives us:

\[\begin{align} \mathbf{A}\mathbf{Q} = \big[\mathbf{A}\mathbf{v}_1, \mathbf{A}\mathbf{v}_2, \ldots, \mathbf{A}\mathbf{v}_n \big]. \label{eq:AQ} \end{align}\]

If we compare Eq. \eqref{eq:AQ} with Eq. \eqref{eq:eigvalues}, we can see that:

\[\begin{align} \mathbf{A}\mathbf{Q} = \big[\lambda_1\mathbf{v}_1, \lambda_2\mathbf{v}_2, \ldots, \lambda_n\mathbf{v}_n \big]. \label{eq:AQ_2} \end{align}\]

If we now define a diagonal matrix carrying the eigenvalues \(\lambda_i\) as

\[\begin{align} \mathbf{\Lambda} = \begin{bmatrix} \lambda_1 & 0 & \ldots & 0\\ 0 & \lambda_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & \lambda_n \end{bmatrix}, \label{eq:Lambda} \end{align}\]

we see that

\[\begin{align} \mathbf{Q}\mathbf{\Lambda} = \big[\lambda_1\mathbf{v}_1, \lambda_2\mathbf{v}_2, \ldots, \lambda_n\mathbf{v}_n \big] \label{eq:AQ_3} \end{align}\]

which is equal to Eq. \eqref{eq:AQ_2}:

\[\begin{align} \mathbf{Q}\mathbf{\Lambda} = \mathbf{A}\mathbf{Q}. \label{eq:AQ_4} \end{align}\]

One final rearrangement – post-multiplying Eq. \eqref{eq:AQ_4} with \(\mathbf{Q}^{-1}\) – and we are done:

\[\begin{align} \mathbf{A} = \mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1}. \label{eq:eigendecomposition} \end{align}\]

Eq. \eqref{eq:eigendecomposition} is also called the eigendecomposition of matrix \(\mathbf{A}\).