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The eigen values of a skew-symmetric matrix are
- a)Always zero
- b)always pure imaginary
- c)Either zero or pure imaginary
- d)always real
Correct answer is option 'C'. Can you explain this answer?
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The eigen values of a skew-symmetric matrix area)Always zerob)always p...
ANSWER :- c
Solution :- Let A be real skew symmetric and suppose λ∈C is an eigenvalue, with (complex) eigenvector v . Then, denoting by H hermitian transposition,
λvHv=vH(λv)=vH(Av)=vH(−AHv)=−(vHAH)v=−(Av)Hv=−(λv)Hv=−λ¯vHv
Since vHv≠0 , as v≠0 , we get
λ=−λ¯
so λ is purely imaginary (which includes 0). Note that the proof works the same for a antihermitian (complex) matrix.
With a completely similar technique you can prove that the eigenvalues of a Hermitian matrix (which includes real symmetric) are real.
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The eigen values of a skew-symmetric matrix area)Always zerob)always p...
Skew-symmetric matrix:
A skew-symmetric matrix is a square matrix in which the transpose of the matrix is equal to the negation of the matrix itself. In other words, the matrix satisfies the condition A^T = -A.
Eigenvalues and Eigenvectors:
Eigenvalues and eigenvectors are important concepts in linear algebra. An eigenvalue of a matrix A is a scalar λ such that when A is multiplied by a corresponding eigenvector v, the result is a scaled version of v. Mathematically, it can be represented as Av = λv.
Eigenvalues of a Skew-Symmetric Matrix:
The eigenvalues of a skew-symmetric matrix have some distinct properties:
1. Imaginary Eigenvalues:
- The eigenvalues of a skew-symmetric matrix are always either zero or pure imaginary.
- This means that the real part of the eigenvalues is always zero.
- The pure imaginary eigenvalues are of the form bi, where b is a non-zero real number.
- For example, if the eigenvalue is 3i, then its conjugate is -3i.
2. Proof:
- Let A be a skew-symmetric matrix and λ be an eigenvalue of A.
- By definition, Av = λv.
- Taking the transpose of both sides, we get v^T A^T = λv^T.
- Since A is skew-symmetric, A^T = -A.
- Substituting this into the equation, we have v^T(-A) = λv^T.
- Rearranging the terms, we get v^T A = -λv^T.
- Taking the transpose again, we have (v^T A)^T = (-λv^T)^T.
- Simplifying, we get Av^T = -λv.
- Therefore, if λ is an eigenvalue of A, then -λ is also an eigenvalue of A.
- This implies that the eigenvalues come in conjugate pairs.
3. Zero Eigenvalues:
- In addition to the pure imaginary eigenvalues, a skew-symmetric matrix can also have zero eigenvalues.
- This occurs when the matrix has at least one row or column of zeros.
- In this case, the eigenvector corresponding to the zero eigenvalue is the zero vector.
Conclusion:
The eigenvalues of a skew-symmetric matrix are either zero or pure imaginary. This is because the transpose of a skew-symmetric matrix is equal to the negation of the matrix itself. The proof shows that if λ is an eigenvalue, then -λ is also an eigenvalue, which implies that the eigenvalues come in conjugate pairs. Additionally, a skew-symmetric matrix can have zero eigenvalues when it has at least one row or column of zeros.
A skew-symmetric matrix is a square matrix in which the transpose of the matrix is equal to the negation of the matrix itself. In other words, the matrix satisfies the condition A^T = -A.
Eigenvalues and Eigenvectors:
Eigenvalues and eigenvectors are important concepts in linear algebra. An eigenvalue of a matrix A is a scalar λ such that when A is multiplied by a corresponding eigenvector v, the result is a scaled version of v. Mathematically, it can be represented as Av = λv.
Eigenvalues of a Skew-Symmetric Matrix:
The eigenvalues of a skew-symmetric matrix have some distinct properties:
1. Imaginary Eigenvalues:
- The eigenvalues of a skew-symmetric matrix are always either zero or pure imaginary.
- This means that the real part of the eigenvalues is always zero.
- The pure imaginary eigenvalues are of the form bi, where b is a non-zero real number.
- For example, if the eigenvalue is 3i, then its conjugate is -3i.
2. Proof:
- Let A be a skew-symmetric matrix and λ be an eigenvalue of A.
- By definition, Av = λv.
- Taking the transpose of both sides, we get v^T A^T = λv^T.
- Since A is skew-symmetric, A^T = -A.
- Substituting this into the equation, we have v^T(-A) = λv^T.
- Rearranging the terms, we get v^T A = -λv^T.
- Taking the transpose again, we have (v^T A)^T = (-λv^T)^T.
- Simplifying, we get Av^T = -λv.
- Therefore, if λ is an eigenvalue of A, then -λ is also an eigenvalue of A.
- This implies that the eigenvalues come in conjugate pairs.
3. Zero Eigenvalues:
- In addition to the pure imaginary eigenvalues, a skew-symmetric matrix can also have zero eigenvalues.
- This occurs when the matrix has at least one row or column of zeros.
- In this case, the eigenvector corresponding to the zero eigenvalue is the zero vector.
Conclusion:
The eigenvalues of a skew-symmetric matrix are either zero or pure imaginary. This is because the transpose of a skew-symmetric matrix is equal to the negation of the matrix itself. The proof shows that if λ is an eigenvalue, then -λ is also an eigenvalue, which implies that the eigenvalues come in conjugate pairs. Additionally, a skew-symmetric matrix can have zero eigenvalues when it has at least one row or column of zeros.
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The eigen values of a skew-symmetric matrix area)Always zerob)always p...
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