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The diagonal elements of a Skew Hermitian Matrix are
- a)All non — zero reals
- b)All purely imaginary number or zero
- c)Some non — zero reals and some imaginary
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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The diagonal elements of a Skew Hermitian Matrix area)All non — ...
Understanding Skew Hermitian Matrices
A Skew Hermitian matrix is defined by the property that its conjugate transpose is equal to its negative. In mathematical terms, for a matrix A, it holds that A* = -A, where A* is the conjugate transpose of A.
Diagonal Elements of Skew Hermitian Matrices
The diagonal elements of a Skew Hermitian matrix have special characteristics:
- Conjugate Property: For a matrix element a_ii (the diagonal element), the property implies that a_ii = -conjugate(a_ii). Since the conjugate of a real number is itself and for an imaginary number i*b (where b is real), the conjugate is -i*b, we can analyze the implications.
- Purely Imaginary or Zero: The only solution to a_ii = -a_ii is that a_ii must be purely imaginary or zero. If a_ii were a non-zero real number, it would lead to a contradiction since it cannot equal its own negative.
Conclusion
From the above properties, we can conclude that:
- The diagonal elements of a Skew Hermitian matrix can only be purely imaginary numbers or zero.
Thus, the correct answer is option 'B': All purely imaginary numbers or zero. This property is crucial in various applications, including quantum mechanics and certain areas of linear algebra.
A Skew Hermitian matrix is defined by the property that its conjugate transpose is equal to its negative. In mathematical terms, for a matrix A, it holds that A* = -A, where A* is the conjugate transpose of A.
Diagonal Elements of Skew Hermitian Matrices
The diagonal elements of a Skew Hermitian matrix have special characteristics:
- Conjugate Property: For a matrix element a_ii (the diagonal element), the property implies that a_ii = -conjugate(a_ii). Since the conjugate of a real number is itself and for an imaginary number i*b (where b is real), the conjugate is -i*b, we can analyze the implications.
- Purely Imaginary or Zero: The only solution to a_ii = -a_ii is that a_ii must be purely imaginary or zero. If a_ii were a non-zero real number, it would lead to a contradiction since it cannot equal its own negative.
Conclusion
From the above properties, we can conclude that:
- The diagonal elements of a Skew Hermitian matrix can only be purely imaginary numbers or zero.
Thus, the correct answer is option 'B': All purely imaginary numbers or zero. This property is crucial in various applications, including quantum mechanics and certain areas of linear algebra.
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