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Q43E

Expert-verifiedFound in: Page 325

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Consider the linear space ${\mathit{V}}$**** of all ${\mathit{n}}{\mathbf{\times}}{\mathit{n}}$**** matrices for which all the vectors ${\stackrel{\mathbf{\rightharpoonup}}{\mathbf{e}}}_{{\mathbf{1}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{\stackrel{\mathbf{\rightharpoonup}}{\mathbf{e}}}_{{\mathbf{n}}}$**** are eigenvectors. Describe the space ${\mathit{V}}$**** (the matrices in ${\mathit{V}}$**** "have a name"), and determine the dimension of ${\mathit{V}}$****.**

Hence, the required dimension is n .

**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 $n\times n$ matrices which all the vectors role="math" localid="1659528394250" ${\stackrel{\rightharpoonup}{\mathrm{e}}}_{\mathrm{b}},{\stackrel{\rightharpoonup}{\mathrm{e}}}_{\mathrm{n}}$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 ${\stackrel{\rightharpoonup}{\mathrm{e}}}_{\mathrm{b}},{\stackrel{\rightharpoonup}{\mathrm{e}}}_{\mathrm{n}}$are called diagonal matrices.

If you put all the vectors ${\stackrel{\rightharpoonup}{\mathrm{e}}}_{z},{\stackrel{\rightharpoonup}{\mathrm{e}}}_{\mathrm{n}}$together in one large $n\times n$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, $dimV=n$.

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