Question

Let A be an symmetric matrix. Assume that A has two different eigenvalues ?1 ?= ?2....

Let A be an symmetric matrix. Assume that A has two different eigenvalues ?1 ?= ?2. Let v1 be a ?1-eigenvector, and v2 be and ?2-eigenvector. Show that v1 ? v2. (Hint: v1T Av2 = v2T Av1.)

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Answer #1

For any real matrix A and any vectors and , we have

Now assume that A is symmetric, and and are eigenvectors of A corresponding to distinct eigenvalues and . Then

Therefore, . Since, , then , i.e.,

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