Label the following statements as true or false. Assume that the underlying inner product spaces are finite-dimensional.(a) Every self-adjoint operator is normal.(b) Operators and their adjoints have the same eigenvectors.(c) If T is an operator on an inner product space V, then T is normal if and only if [T]β is normal, where β is any ordered basis for V.(d) A real or complex matrix A is normal if and only if LA is normal.(e) The eigenvalues of a self-adjoint operator must all be real.(f) The identity and zero operators are self-adjoint.(g) Every normal operator is diagonalizable.(h) Every self-adjoint operator is diagonalizable

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Asked Dec 30, 2019
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Label the following statements as true or false. Assume that the underlying inner product spaces are finite-dimensional.

(a) Every self-adjoint operator is normal.

(b) Operators and their adjoints have the same eigenvectors.

(c) If T is an operator on an inner product space V, then T is normal if and only if [T]β is normal, where β is any ordered basis for V.

(d) A real or complex matrix A is normal if and only if LA is normal.

(e) The eigenvalues of a self-adjoint operator must all be real.(f) The identity and zero operators are self-adjoint.

(g) Every normal operator is diagonalizable.

(h) Every self-adjoint operator is diagonalizable

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Expert Answer

Step 1

(a) True.

If T = T*, then T*T = T2 = TT*.

Advanced Math homework question answer, step 1, image 1
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Step 2

(c) False.

The statement requires beta being on an orthonormal basis. The given statement is not true for every ordered basis.

Advanced Math homework question answer, step 2, image 1
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Step 3
Advanced Math homework question answer, step 3, image 1
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