Dimensionality Reduction - Mining of Massive Datasets - page 390

11.1.4 The Matrix of Eigenvectors

Suppose we have an n × n matrix M whose eigenvectors, viewed as column vectors, are e 1 ,
e 2 , . . . , e n . Let E be the matrix whose i th column is e i . Then EE T = E T E = I . The explana-
tion is that the eigenvectors of a matrix are orthonormal . That is, they are orthogonal unit
vectors.

EXAMPLE 11.5 For the matrix M of Example 11.2 , the matrix E is

E T is therefore

When we compute EE T we get

The calculation is similar when we compute E T E . Notice that the 1's along the main di-
agonal are the sums of the squares of the components of each of the eigenvectors, which
makes sense because they are unit vectors. The 0's off the diagonal reflect the fact that the
entry in the i th row and j th column is the dot product of the i th and j th eigenvectors. Since
eigenvectors are orthogonal, these dot products are 0.

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11.1.5 Exercises for Section 11.1

EXERCISE 11.1.1 Find the unit vector in the same direction as the vector [ 1 , 2 , 3 ] .

EXERCISE 11.1.2 Complete Example 11.4 by computing the principal eigenvector of the
matrix that was constructed in this example. How close to the correct solution (from
Example 11.2 ) are you?

EXERCISE 11.1.3 For any symmetric 3 × 3 matrix

there is a cubic equation in λ that says the determinant of this matrix is 0. In terms of a
through f , find this equation.