A symmetric matrix A has eigenvalues 1 and 2. Find A if 1 1 1 spans E(2)

Chapter 6, Problem 3

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QUESTION:

A symmetric matrix A has eigenvalues 1 and 2. Find A if \(\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]\) spans E(2).

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QUESTION:

A symmetric matrix A has eigenvalues 1 and 2. Find A if \(\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]\) spans E(2).

ANSWER:

Step 1 of 2

Consider  and . Since, any symmetric matrix is diagonalizable, the algebraic and geometric multiplicities of all eigenvalues is equal.

Thus, if  is the geometric multiplicity of  and  is the algebraic multiplicity of ,we get  and  

Since,  and there are only two eigenvalues.

Thus,  is a two-dimensional plane in .

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