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Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option a:<\/strong> IF \\( A=UDU^* \\) with \\( U \\) unitary and \\( D \\) diagonal, the first off \\( D \\) must be invertible (why?) and so \\( A^{-1} = (UDU^*)^{-1} = (U^*)^{-1} D^{-1} U^{-1} = U D^{-1} U^* \\) since \\( U \\) is unitary. So \\( A^{-1} \\) is unitarily similar (via the same unitary matrix \\( U \\)) to the diagonal matrix \\( D^{-1} \\) (why is \\( D^{-1} \\) diagonal?).<\/div>\n","notChosenFeedback":"
Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option a:<\/strong> IF \\( A=UDU^* \\) with \\( U \\) unitary and \\( D \\) diagonal, the first off \\( D \\) must be invertible (why?) and so \\( A^{-1} = (UDU^*)^{-1} = (U^*)^{-1} D^{-1} U^{-1} = U D^{-1} U^* \\) since \\( U \\) is unitary. So \\( A^{-1} \\) is unitarily similar (via the same unitary matrix \\( U \\)) to the diagonal matrix \\( D^{-1} \\) (why is \\( D^{-1} \\) diagonal?).<\/div>\n","tip":""},"text":"
a.<\/strong> If \\( A \\) is unitarily diagonalizable and invertible, then \\( A^{-1} \\) is unitarily diagonalizable.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"
Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option b:<\/strong> Counterexample: The \\( 2\\times2 \\) zero matrix is certainly unitarily diagonalizable (it's already diagonal). Every non-zero vector is an eigenvector, so just choose any two non-orthogonal, non-zero vectors to get a counterexample.<\/div>\n","notChosenFeedback":"
Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option b:<\/strong> Counterexample: The \\( 2\\times2 \\) zero matrix is certainly unitarily diagonalizable (it's already diagonal). Every non-zero vector is an eigenvector, so just choose any two non-orthogonal, non-zero vectors to get a counterexample.<\/div>\n","tip":""},"text":"
b. <\/strong>If \\( A \\) is unitarily diagonalizable, and if \\( \\vec{x} \\) and \\( \\vec{y} \\) are distinct eigenvectors of \\( A \\), then \\( \\vec{x} \\) and \\( \\vec{y} \\) are orthogonal (with respect to the standard inner product on \\( \\mathbb{C}^n \\)).<\/div>\n"},{"correct":true,"tipsAndFeedback":{"chosenFeedback":"
Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option c:<\/strong> This is the fundamental property of the adjoint.<\/div>\n","notChosenFeedback":"
Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option c:<\/strong> This is the fundamental property of the adjoint.<\/div>\n","tip":""},"text":"
c.<\/strong> For all \\( \\vec{x},\\vec{y}\\in\\mathbb{C}^n \\), \\( \\langle A\\vec{x}, \\vec{y} \\rangle = \\langle \\vec{x}, A^*\\vec{y} \\rangle \\).<\/div>\n"},{"correct":false,"tipsAndFeedback":{"chosenFeedback":"
Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option d:<\/strong> Counterexample: \\( A=\\begin{bmatrix} 0&1\\\\0&0 \\end{bmatrix} \\). Then \\( A^*= \\begin{bmatrix} 0&0\\\\1&0 \\end{bmatrix} \\). Observe that \\( \\begin{bmatrix} 1\\\\0 \\end{bmatrix} \\) is in \\( \\text{Null}(A) \\) but not in \\( \\text{Null}(A^*) \\).<\/div>\n","tip":"","notChosenFeedback":"
Answer: <\/strong>The correct choices are a and c.<\/div>\n\n
Reasoning for option d:<\/strong> Counterexample: \\( A=\\begin{bmatrix} 0&1\\\\0&0 \\end{bmatrix} \\). Then \\( A^*= \\begin{bmatrix} 0&0\\\\1&0 \\end{bmatrix} \\). Observe that \\( \\begin{bmatrix} 1\\\\0 \\end{bmatrix} \\) is in \\( \\text{Null}(A) \\) but not in \\( \\text{Null}(A^*) \\).<\/div>\n"},"text":"
d.<\/strong> \\( \\text{Null}(A^*)=\\text{Null}(A) \\).<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":true,"enableSolutionsButton":false,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":false,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"Check","submitAnswerButton":"Submit","showSolutionButton":"Show solution","tryAgainButton":"Retry","tipsLabel":"Show tip","scoreBarLabel":"You got :num out of :total points","tipAvailable":"Tip available","feedbackAvailable":"Feedback available","readFeedback":"Read feedback","wrongAnswer":"Wrong answer","correctAnswer":"Correct answer","shouldCheck":"Should have been checked","shouldNotCheck":"Should not have been checked","noInput":"Please answer before viewing the solution","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"
Select all true statements concerning the matrix \\( A\\in M_{n\\times n}(\\mathbb{C}) \\).<\/p>\n"}