Question

In Exercises 25 and 26, mark each statement True or False. Justify each answer.

26.

1. There are symmetric matrices that are not orthogonally diagonizable.
2. b. If \(B = PD{P^T}\), where \({P^T} = {P^{ - {\bf{1}}}}\) and D is a diagonal matrix, then B is a symmetric matrix.
3. c. An orthogonal matrix is orthogonally diagonizable.
4. d. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

Short answer

(a) The statement is False.

(b) The statement is True.

(c) The statement is False.

(d) The statement isFalse.

Step-by-step solution

Find an answer for part (a)

According to theorem 2, a matrix of order \(n \times n\) is orthogonally diagonalizable when the matrix is symmetric.

So, the statement is False.

Find an answer for part (b)

As \({P^T} = {P^{ - 1}}\), then P is an orthogonal matrix, so by the equation,\(B = PD{P^T}\) it shows thatB is orthogonally diagonalizable, and thus,B is a symmetric matrix.

So, the statement is True.

Find an answer for part (c)

An orthogonal matrix can be symmetric, but not every orthogonal matrix, but not all orthogonal matrix is symmetric.

Thus, the statement is False.

Find an answer for part (d)

According to theorem 3(b), the dimension of eigenspace is less than or equal to the multiplicity corresponding to the eigenvalue. But it can be less than the value of multiplicity of the corresponding eigenvalue for a symmetric matrix.

Thus, the statement is False.