Question

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

a. An\(n \times n\)matrix that is orthogonally diagonalizable must be symmetric.

b. If\({A^T} = A\)and if vectors\({\rm{u}}\)and\({\rm{v}}\)satisfy\(A{\rm{u}} = {\rm{3u}}\)and\(A{\rm{v}} = {\rm{3v}}\), then\({\rm{u}} \cdot {\rm{v}} = {\rm{0}}\).

c. An\(n \times n\)symmetric matrix has n distinct real eigenvalues.

d. For a nonzero \({\rm{v}}\) in \({\mathbb{R}^n}\) , the matrix \({\rm{v}}{{\rm{v}}^T}\) is called a projection matrix.

Short answer

1. The given statement is True.
2. The given statement is True.
3. The given statement is False.
4. The given statement is False.

Step-by-step solution

Apply theorem 2

According to theorem 2, an\(n \times n\)matrix\(A\)is orthogonally diagonalizable if it is symmetric.

So, statement (a) is true.

Interpret theorem 2

According to another interpretation of theorem 2, an\(n \times n\)symmetric matrix\(A\), having the non-zero eigenvectors\({\rm{u,}}\,{\rm{v}}\),is orthogonally diagonalizable.

Thus, \(\lambda = 3\)must be one of the eigenvalues of matrix\(A\).

So, the characteristic equation\(A{\rm{u}} = 3{\rm{u}}\)and\(A{\rm{v}} = 3{\rm{v}}\)must be true.

As a result, statement in (b) is true.

Apply theorem 3

According to theorem 3, an\(n \times n\)symmetric matrix\(A\)have \(n\) eigenvalues, and may have counting multiplicities.

Thus, eigenvalues need not to be distinct. So, the statement in (c) is false.

Apply the concept of projection matrix

If \({\rm{u}}\)is a unit vector, then ,\(\lambda {{\rm{u}}_j}{{\rm{u}}_j}^T\)is a projection matrix such that for each vector\({\rm{v}}\)in\({\mathbb{R}^n}\),\(\left( {{\rm{u}}{{\rm{u}}^T}} \right){\rm{v}}\)is an orthogonal projection of\({\rm{v}}\)in the subspace spanned by\({\rm{u}}\).

So, the statement in (d) is false.