Question about normal matrices

Thread starter chuy52506
Start date Apr 3, 2011
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Matrices Normal

In summary, a normal matrix is a square matrix that commutes with its conjugate transpose. It can be determined if a matrix is normal by checking if it commutes with its conjugate transpose. Normal matrices have important properties, such as being diagonalizable by a unitary matrix and having real eigenvalues. Non-square matrices cannot be normal. Normal matrices are used in various applications, including signal processing, quantum mechanics, and statistics, to represent complex systems and improve efficiency in calculations.

Apr 3, 2011

chuy52506

Homework Statement

Suppose A is a normal matrix in the complex field.

Homework Equations

Show that ||Ax||=||A^*x|| for all x in the complex field

The Attempt at a Solution

If A is normal then AA^*=A^*A and ||Ax||=(Ax,Ax)=(x,A^*Ax)

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Apr 3, 2011

chuy52506

Never mind i got it=]

1. What is a normal matrix?

A normal matrix is a square matrix that commutes with its conjugate transpose. In other words, the matrix multiplied by its conjugate transpose is equal to the conjugate transpose multiplied by the matrix.

2. How can I determine if a matrix is normal?

A matrix is normal if and only if it commutes with its conjugate transpose. This means that if the matrix is multiplied by its conjugate transpose and the result is equal to the conjugate transpose multiplied by the matrix, then the matrix is normal.

3. What are the properties of normal matrices?

Normal matrices have several important properties, including the fact that they are diagonalizable by a unitary matrix, they have orthogonal eigenvectors, and they have real eigenvalues.

4. Can a non-square matrix be normal?

No, a non-square matrix cannot be normal. Normality is a property that only applies to square matrices.

5. How are normal matrices used in applications?

Normal matrices have many important applications in fields such as signal processing, quantum mechanics, and statistics. They are used to represent complex systems and make calculations more efficient, among other things.