Application of Eigenvectors

Wednesday, 5 September 2018

2:27 PM

Let 𝑉 be an 𝑛-dimensional vector space and 𝑇:𝑉→𝑉 be a linear transformation. If 𝑇 has 𝑛 linearly independent eigenvectors, they form a basis for 𝑉. This basis is "tailor made" for 𝑇.

For a linear map 𝑇:𝑉→𝑉, suppose that the 𝑛 eigenvectors ﷐𝑣﷮1﷯, …,﷐𝑣﷮𝑛﷯ of 𝑇 form a basis for 𝑉 and the igenvalues are eigenvalues are respectively ﷐λ﷮1﷯,…,﷐λ﷮𝑛﷯. Find 𝑇(﷐𝑘﷮1﷯﷐𝑣﷮1﷯+…+﷐𝑘﷮𝑛﷯﷐𝑣﷮𝑛﷯) as a linear combination of eigenvectors.

If an 𝑛×𝑛 matrix has 𝑛 distinct eigenvalues, then it has 𝑛 linearly independent eigenvectors

Suppose that 𝐴 is a 3×3 matrix with eigenvalues 1, 3, -2 and corresponding the eigenvectors ﷐𝑣﷮1﷯, ﷐𝑣﷮2﷯, ﷐𝑣﷮3﷯. Explain why the eigenvectors are linearly indepndent.
-> Since the 3 eigenvalues are distinct, the corresponding eigenvectors are linearly independent.

Find ﷐𝐴﷮2﷯(2﷐𝑣﷮1﷯−﷐𝑣﷮2﷯−2﷐𝑣﷮3﷯) and ﷐𝐴﷮𝑛﷯﷐﷐2﷮𝑣1﷯−﷐𝑣﷮2﷯−2﷐𝑣﷮3﷯﷯

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