91Ó°ÊÓ

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

Let’s consider the linear space $V$or all $n×n$matrices which all the vectors role="math" localid="1659528394250" ${\stackrel{â‡\P Ä}{\mathrm{e}}}_{\mathrm{b}},{\stackrel{â‡\P Ä}{\mathrm{e}}}_{\mathrm{n}}$are Eigen vectors.

We want to describe the space and determine its dimension.

The set of matrices in the space $V$spanned by the Eigen vectors ${\stackrel{â‡\P Ä}{\mathrm{e}}}_{\mathrm{b}},{\stackrel{â‡\P Ä}{\mathrm{e}}}_{\mathrm{n}}$are called diagonal matrices.

If you put all the vectors ${\stackrel{â‡\P Ä}{\mathrm{e}}}_z,{\stackrel{â‡\P Ä}{\mathrm{e}}}_{\mathrm{n}}$together in one large $n×n$matrix it will have only values across the diagonal, thus the name diagonal matrix.

Since all $n$Eigen vectors span the space and are linearly independent.

Therefore, $dimV=n$.