Solving Linear Systems with Hermitian Matrices

In summary, the student was able to find their eigenvalues and they were both real as expected. The first eigenvalue was -3 which they knew was correct. The equations they were given were 5x+(3-i)y=0 and (3+i)x+2y=0, and they came from a hermitian matrix after substituting in a value of λ=-3. They struggled with solving this particular problem but found it to be solvable by trial and error. They also questioned how they would proceed with harder equations, but eventually figured out that they only needed to set ax = by and force a value for either x or y.
  • #1
tamtam402
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Homework Statement


I can find my eigenvalues just fine, and they're both real, as expected. My first eigenvalue is -3, which I know is correct.

I have the equations 5x+(3-i)y=0, (3+i)x+2y=0

Both of the equations come from my hermitian matrix, after I substituted λ=-3.

Homework Equations





The Attempt at a Solution



I have absolutely no idea how to solve this. This case is simple enough to be solved by trial and error, but how would I proceed if I had harder equations?

I can't use both equations since I get x=x or y=y if I substitute one into the other, since they're both the same equation.
 
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  • #2


Nevermind, doing another (easier) exercise allowed me to see that I only have to set ax = by, and force a value to either x or y.
 

1. What is a Hermitian matrix?

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, the matrix is equal to its complex conjugate reflected over the main diagonal. This means that all entries on the main diagonal are real numbers, and the entries above and below the main diagonal are complex conjugates of each other.

2. How do you solve linear systems with Hermitian matrices?

To solve a linear system with Hermitian matrices, we can use the Cholesky decomposition method. This involves factoring the Hermitian matrix into the product of a lower triangular matrix and its conjugate transpose. Then, we can use this factorization to solve the system of equations.

3. What are some properties of Hermitian matrices?

Some properties of Hermitian matrices include:

  • They have real eigenvalues.
  • Their eigenvectors corresponding to distinct eigenvalues are orthogonal.
  • They are always diagonalizable.
  • Their determinant is always a real number.
  • Their inverse is also a Hermitian matrix.

4. How are Hermitian matrices used in quantum mechanics?

In quantum mechanics, Hermitian matrices are used to represent observable quantities, such as position, momentum, and energy. The eigenvalues of these matrices correspond to the possible outcomes of a measurement, and the eigenvectors represent the states of the system.

5. Can non-square matrices be Hermitian?

No, non-square matrices cannot be Hermitian. By definition, a Hermitian matrix must be a square matrix with complex entries. Non-square matrices do not have a conjugate transpose, which is necessary for a matrix to be Hermitian.

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