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Linear Algebra With Applications
Found in: Page 325
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Consider the linear space V of all n×n matrices for which all the vectors e1,.....,en are eigenvectors. Describe the space V (the matrices in V "have a name"), and determine the dimension of V.

Hence, the required dimension is n .

See the step by step solution

Step by Step Solution

Step 1: Definition of the Eigenvectors

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.

Step 2: Find dimension

Let’s consider the linear space Vor all n×n matrices which all the vectors role="math" localid="1659528394250" eb,enare 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 eb,enare called diagonal matrices.

If you put all the vectors ez,entogether in one large n×nmatrix 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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