Is Q^{-1}AQ^{-1} Always Hermitian?

td21 · 2011-06-13T07:14:57+00:00

Hi! Q is postive definite A is any matrix. Why Q^{-1}AQ^{-1} is hermitian??

Thread starter td21
Start date Jun 13, 2011
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Hermitian Matrix

In summary, a hermitian matrix is a square matrix that is equal to its own conjugate transpose. It is important for its many properties and applications in mathematics and physics, and can be determined by checking if it satisfies the equation A = A<sup>*</sup>. The eigenvalues of a hermitian matrix are always real numbers, and a non-square matrix cannot be hermitian.

Jun 13, 2011

td21

Gold Member

Hi!

Q is postive definite

A is any matrix.

Why [itex] Q^{-1}AQ^{-1} [/itex] is hermitian??

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Jun 13, 2011

dragonlorder

let Q be identity matrix, and A be a non symmetric matrix
Q is positive definite
but the product of those 3 is not hermitian

Related to Is Q^{-1}AQ^{-1} Always Hermitian?

1. What does it mean for a matrix to be hermitian?

A hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that the elements along the main diagonal are real numbers, and the elements above and below the diagonal are complex conjugates of each other.

2. Why is it important for a matrix to be hermitian?

Hermitian matrices have many important properties and applications in mathematics and physics. They are used in quantum mechanics, signal processing, and optimization problems. They also have many nice properties that make them easier to work with in calculations.

3. How can I determine if a matrix is hermitian?

A matrix can be tested for hermitian properties by checking if it is equal to its own conjugate transpose. This means that the matrix A must satisfy the equation A = A^*, where A^* is the conjugate transpose of A. If this equation is satisfied, then the matrix is hermitian.

4. What are the eigenvalues of a hermitian matrix?

The eigenvalues of a hermitian matrix are always real numbers. This is because the eigenvalues are the roots of the characteristic polynomial of the matrix, and for hermitian matrices, this polynomial has only real coefficients.

5. Can a non-square matrix be hermitian?

No, a non-square matrix cannot be hermitian. Hermitian matrices must be square, meaning they have the same number of rows and columns. This is because the conjugate transpose operation only applies to square matrices.