# Expressing each column vector of a matrix as a linear combination of Eigen vectors of that matrix

Let's say we have a full rank square matrix $$A$$ of size $$n$$ and we have found the eigenvectors of the matrix through eigen-decomposition/SVD.

Here, $$A = [ V_1 , V_2 , V_3 , \ldots, V_n ]$$ , where $$V_i$$ is a vector of size $$n\times 1$$.

How can I express $$V_i$$ as a linear combination of eigenvectors of $$A$$?

## 1 Answer

If $$A$$ ist normal then by the spectral theorem $$A=UDU^T$$ where $$D$$ is a diagonal matrix containing the eigenvalues and $$U$$ is orthonormal matrix whose columns are the eigenvectors.

Then all you have to do is solve the linear system $$UX=A$$ since then then the first column if $$A$$ equals a linear combination of the columns of $$U$$ whose coefficients are given by the first column of $$X$$, etc.

Since $$U$$ is orthonormal, the system is trivial to solve: $$X= U^T A=DU^T$$. So the Eigendecomposition already encodes this information.