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Chapter 7: Q7-20E (page 383)

TRUE OR FALSE

20. The matrix of any orthogonal projection onto a subspace V of Rn is diagonalizable.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Define eigenvalues

The scalar values that are associated with the vectors of the linear equations in the matrix are called eigenvalues.

Ax鈫�=位x鈫�, herex鈫�is eigenvector and位is the eigenvalue.

02

Determine whether the given statement is true or false

Hence, the orthogonal projection matrix always has two eigenvalues,位1=1,withE1=Vand位2=0with E0=V鈯�.

Thus, we have dimE1+dimE0=dimV+dimV鈯�=n, so this matrix is always diagonalizable.

Therefore, the given statement is true.

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