Chapter 8: Problem 20
If \(A\) is hermitian, show that all the coefficients of \(c_{A}(x)\) are real numbers.
Short Answer
Expert verified
For a hermitian matrix \(A\), the characteristic polynomial \(c_A(x)\) has all real coefficients because it has real eigenvalues.
Step by step solution
01
Understanding Hermitian Matrix
A hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. This means that for matrix \(A\), \(A = A^*\) where \(^*\) denotes the conjugate transpose of \(A\). In simpler terms, for a hermitian matrix, the element \(a_{ij} = \overline{a_{ji}}\).
02
Understanding Characteristic Polynomial
The characteristic polynomial \(c_A(x)\) of a matrix \(A\) is given by \(\det(xI - A)\), where \(I\) is the identity matrix of the same dimension as \(A\). The coefficients of this polynomial are derived from the determinant of \(xI - A\).
03
Real Coefficients for Hermitian Matrices
A fundamental property of hermitian matrices is that they have real eigenvalues. Since \(c_A(x)\) is a characteristic polynomial, its roots are the eigenvalues of \(A\). Thus, all roots are real numbers.
04
Link between Roots and Coefficients
The coefficients of a polynomial are determined by the roots through Viète's formulas, so if all the roots (eigenvalues) of the polynomial are real, the coefficients must also be real. This is because complex coefficients would require the existence of complex roots, contradicting the fact that the eigenvalues are real.
05
Conclusion on Hermitian Matrices
Since the eigenvalues (roots of the characteristic polynomial) of a hermitian matrix are guaranteed to be real, the characteristic polynomial \(c_A(x)\) must have all real coefficients.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Characteristic Polynomial
The characteristic polynomial of a matrix is a fascinating concept in linear algebra. It functions as a key to unlocking the properties of the matrix itself. To determine the characteristic polynomial of a matrix \(A\), we use the formula \(c_A(x) = \det(xI - A)\). Here, \(I\) stands for the identity matrix of the same dimension as \(A\). This polynomial is crucial because its roots are the eigenvalues of the matrix, which are essential in various fields such as quantum mechanics or vibration analysis.
- The coefficients of the characteristic polynomial provide insights into the matrix's behavior.
- The polynomial's degree equals the dimension of the matrix.
Real Eigenvalues
Real eigenvalues are particularly significant in the context of Hermitian matrices. An eigenvalue is essentially a scalar that signifies how much the matrix stretches or shrinks a vector during a transformation. For Hermitian matrices, which are prevalent in quantum mechanics and other scientific disciplines, all eigenvalues are real numbers. This property comes from the definition of a Hermitian matrix, which is equal to its own conjugate transpose.
- A matrix is Hermitian if for any elements \(a_{ij}\) of the matrix, \(a_{ij} = \overline{a_{ji}}\).
- Because the eigenvalues are real, they simplify many calculations in physics and engineering.
Complex Matrices
Complex matrices are an extension of regular numeric matrices and are widely used, especially in disciplines where phase and magnitude play a critical role, like electrical engineering and quantum physics. A complex matrix consists of complex numbers, which have both a real part and an imaginary part. When examining the Hermitian matrix as a specific type of complex matrix, certain advantageous properties emerge. For instance, Hermitian matrices, while having complex entries, still maintain useful characteristics such as having real eigenvalues.
- If \(A\) is a Hermitian matrix, then \(A = A^*\), where \(^*\) denotes the conjugate transpose.
- Complex matrices benefit from operations such as conjugation and transposition, providing tools to illuminate solutions to quantum mechanics problems.