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A symmetric matrix A has eigenvalues 1 and 2. Find A if 1 1 1 spans E(2)
Chapter 6, Problem 3(choose chapter or problem)
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).
Questions & Answers
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 .