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Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A then
- a)α can be any real number
- b)α = 1 or - 1
- c)α can be any complex number of absolute value 1
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
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Let A be a matrix with complex enteries. If A is hermitian as well as ...
If a matrix A is both Hermitian and unitary, then it must be a diagonal matrix with diagonal entries of absolute value 1.
To prove this, let's start with the definition of a Hermitian matrix. A matrix A is Hermitian if it is equal to its conjugate transpose, i.e., A = A*.
Now, let's consider the unitary property of A. A matrix A is unitary if its conjugate transpose is equal to its inverse, i.e., A* = A^(-1).
Combining these two properties, we have A = A* = A^(-1).
Taking the conjugate transpose of both sides, we get A* = (A^(-1))*.
Now, let's multiply both sides of A = A* = A^(-1) by A to get A^2 = A*A^(-1) = I, where I is the identity matrix.
Taking the conjugate transpose of both sides, we have (A^2)* = (A*A^(-1))* = I* = I.
Now, let's multiply both sides of A^2 = I by A* to get (A^2)*(A*) = I*(A*).
Using the property of matrix multiplication, we have (A*A^(-1))*(A*) = A*(A*A^(-1)) = A*I = A.
Simplifying the left side, we get A*A^2 = A.
Since matrix multiplication is associative, we have A*(A*A) = A.
Using the definition of a Hermitian matrix, we have A*(A*A) = (A*A)* = A^2* = A.
Therefore, A is equal to its conjugate transpose squared, i.e., A = A^2.
This means that A is a diagonal matrix, since the diagonal entries of A^2 are the squares of the diagonal entries of A.
Furthermore, since A is Hermitian, its diagonal entries must be real numbers. Therefore, the diagonal entries of A are real numbers of absolute value 1.
In conclusion, if a matrix A is both Hermitian and unitary, it must be a diagonal matrix with diagonal entries of absolute value 1.
To prove this, let's start with the definition of a Hermitian matrix. A matrix A is Hermitian if it is equal to its conjugate transpose, i.e., A = A*.
Now, let's consider the unitary property of A. A matrix A is unitary if its conjugate transpose is equal to its inverse, i.e., A* = A^(-1).
Combining these two properties, we have A = A* = A^(-1).
Taking the conjugate transpose of both sides, we get A* = (A^(-1))*.
Now, let's multiply both sides of A = A* = A^(-1) by A to get A^2 = A*A^(-1) = I, where I is the identity matrix.
Taking the conjugate transpose of both sides, we have (A^2)* = (A*A^(-1))* = I* = I.
Now, let's multiply both sides of A^2 = I by A* to get (A^2)*(A*) = I*(A*).
Using the property of matrix multiplication, we have (A*A^(-1))*(A*) = A*(A*A^(-1)) = A*I = A.
Simplifying the left side, we get A*A^2 = A.
Since matrix multiplication is associative, we have A*(A*A) = A.
Using the definition of a Hermitian matrix, we have A*(A*A) = (A*A)* = A^2* = A.
Therefore, A is equal to its conjugate transpose squared, i.e., A = A^2.
This means that A is a diagonal matrix, since the diagonal entries of A^2 are the squares of the diagonal entries of A.
Furthermore, since A is Hermitian, its diagonal entries must be real numbers. Therefore, the diagonal entries of A are real numbers of absolute value 1.
In conclusion, if a matrix A is both Hermitian and unitary, it must be a diagonal matrix with diagonal entries of absolute value 1.
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Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
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Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?.
Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?.
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Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of Let A be a matrix with complex enteries. If A is hermitian as well as unitary and α is an eigen values of A thena)α can be any real numberb)α = 1 or - 1c)α can be any complex number of absolute value 1d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
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