Given Information
We are given with some statements, that we have to to prove if they are True or False
Step-1: (a)
An matrix is said to be symmetric if and only if it is orthogonally diagonalizable. Hence, every symmetric matrix can be diagonalizable orthogonally. Therefore
The given statement is False
Step-2: (b)
If P
T
=
P
−
1
, then P is orthonormal matrix: Hence if
B
=
P
D
P
T
Then, B is orthogonally diagonalizable and hence as given in part (a), it is a symmetric matrix:
The given statement is True
Step-3: (c)
A matrix is symmetric if and only if it is orthogonally diagonalizable. Hence, every symmetric matrix is orthogonally diagonalizable. However, not all the orthonormal matrices are symmetric. Therefore
The given statement is False
Step-4: (d)
The dimension of eigenspace is less than or equal to the multiplicity of the corresponding eigenvalue. Also, for the symmetric matrix, the dimension of the eigenspace is sometimes less than the multiplicity of the corresponding eigenvalue.