2. any tips how to do the problem would be appreciated! thanks! (Dimensions of E

Question

2.

any tips how to do the problem would be appreciated! thanks!

(Dimensions of Eigenspaces) Let A = . Find all eigenvalues of A. Find a basis for each eigenspace of A. What is the sum of the dimensions of the eigenspaces of A? Based on your answer to the previous part, guess a formula for the sum of the dimensions of the eigenspaces of a real n times n symmetric matrix. Explain why your formula must work for any real n times n symmetric matrix. Let x1 = , where a2 + b2 + c2 = 1. Find vectors x2 and x3 such that {x1, x2, x3} is an orthonormal basis for R3.

Explanation / Answer

As you can see, this is clearly two questions. I'll address the second. If x1 = (a b c), then define x2 = (1 0 0). The vector x1perp = x2 - (x1.x2) x1 is perpendicular to x1. A third, linearly independent vector is obtained by the cross product between x1 and x1perp. Since the three are linearly independent, they necessarily span R3. To ensure that they are orthonormal, just rescale so that each has norm 1.

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