Linear Operators and Their Properties

Let TT be a linear operator on an inner product space. TT is said to be normal if:

• T=T∗T = T^*

• T∗T=TT∗T^*T = TT^*

• T∗=−TT^* = -T

• TT is diagonalizable

If T is self-adjoint, then:

• All eigenvalues of T are real

• T is not normal

• T has no eigenvalues

• T is skew-symmetric

Which of the following operators is always normal?

• Self-adjoint operator

• Skew-symmetric operator

• Projection operator

• Nilpotent operator

If T is a normal operator on a complex inner product space, then it is:

• Not diagonalizable

• Diagonalizable using an orthonormal basis

• Diagonalizable only if it is unitary

• Always self-adjoint

The matrix of a self-adjoint operator is:

• Skew-Hermitian

• Unitary

• Hermitian

• Orthogonal

The matrix of a unitary operator is:

• Hermitian

• Orthogonal

• Normal

• Skew-Hermitian

Which of the following is not a property of a unitary operator?

• Length-preserving

• Angle-preserving

• T∗T=IT^*T = I

• T∗=−TT^* = -T

Which of the following matrices is always unitary?

• Identity matrix

• Skew-Hermitian matrix

• Zero matrix

• Upper-triangular matrix

Which of the following statements is false?

• All unitary operators are normal

• All self-adjoint operators are unitary

• All unitary matrices are invertible

• A unitary operator preserves n

The eigenvalues of an orthogonal projection are:

• 0 and 1

• All real numbers

• Only 1

• Only 0