Orthogonal transformation of matrix

[vabamyyr]

I have a question on matrix norms and orthogonal transformations. The 2-norm in invariant under orthogonal transformation, for if Q^T*Q=I. But i have trouble showing that for orthogonal Q and Q^H with appropriate dimensions

|| Q^H*A*Q ||2 =|| A ||2

 

[doodle]

The 2-norm of A returns the square-root of the maximum absolute eigenvalues of A^HA. So check, does (Q^HA*Q)^HQ^HA*Q preserve the absolute eigenvalues of A^HA?

 

[tacman]

Orthogonal transformation of a matrix refers to the process of multiplying a matrix by an orthogonal matrix, resulting in a transformed matrix with the same determinant, and preserving the angles and lengths of the vectors within the matrix. This is an important concept in linear algebra and has various applications in fields such as computer graphics, signal processing, and physics.

One interesting property of orthogonal transformations is that they preserve the 2-norm of a matrix. This means that if we have a matrix A and we transform it by multiplying it on both sides by an orthogonal matrix Q, the 2-norm of the transformed matrix Q^H*A*Q will be the same as the 2-norm of the original matrix A.

To show this, we can use the fact that the 2-norm of a matrix is defined as the square root of the maximum eigenvalue of A^H*A. Since Q is an orthogonal matrix, Q^H*Q=I, and thus Q^H*A*Q is similar to A. This means that they have the same eigenvalues and thus the same maximum eigenvalue. Therefore, the 2-norm of Q^H*A*Q is equal to the 2-norm of A.

In other words, the 2-norm is invariant under orthogonal transformation. This is a useful property that allows us to analyze and compare matrices without being affected by their orientation or rotation.

However, it is important to note that this property only holds for orthogonal matrices Q and Q^H with appropriate dimensions. This means that Q and Q^H must be square matrices of the same size. If this condition is not met, the 2-norm may not be preserved under orthogonal transformation.

In conclusion, the 2-norm of a matrix is invariant under orthogonal transformation, which is a useful property in many applications. This can be shown by using the fact that orthogonal matrices preserve the eigenvalues of a matrix, and thus the maximum eigenvalue and 2-norm remain unchanged. However, it is important to ensure that the dimensions of the orthogonal matrices are appropriate to maintain this property.