What is the difference between the two definitions of Schur decomposition?

In summary, the Schur decomposition of a matrix A can be defined as either A = Q U Q^{-1} or Q^H A Q = T, where Q is unitary and U/T are upper triangular matrices. The two definitions may seem different due to the placement of the matrix, but because Q is unitary, they are equivalent. The Schur decomposition essentially represents a change of basis for the matrix A.
  • #1
junglebeast
515
2
Wikipedia defines the Shur decomposition of matrix A as

A = Q U Q^{-1}

where Q is unitary and U is upper triangular.

http://en.wikipedia.org/wiki/Schur_decomposition

Mathworld defines the Shur decomposition of matrix A as

Q^H A Q = T,

where Q is unitary and T is upper triangular.

http://mathworld.wolfram.com/SchurDecomposition.html

Because Q is unitary, the inverse is the same as the conjugate transpose...but they still seem like completely different definitions because the matrix is either on the inside or the outside. What's the truth?
 
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  • #2
Since Q^H = Q^-1, you have

Q Q^H A Q = Q T
=>
Q Q^-1 A Q = Q T
=>
I A Q = Q T
=>
A Q = Q T
=>
A Q Q^-1 = Q T Q^-1
=>
A I = Q T Q^-1
=>
A = Q T Q^-1
 
  • #3
Wow, I feel stupid for not noticing that! Thanks
 
  • #4
A is just a change of basis, I'd recommend reviewing change of basis in lin alg
 

1. What is the Schur decomposition ambiguity?

The Schur decomposition ambiguity refers to the fact that a square matrix can have multiple Schur decompositions, meaning it can be expressed as a product of upper triangular matrices in different ways. This ambiguity arises due to the possibility of rearranging the order of the diagonal elements in the upper triangular matrices without affecting the overall product.

2. How does the Schur decomposition relate to eigenvalues and eigenvectors?

The Schur decomposition is closely related to the eigenvalue decomposition, as both involve diagonalizing a matrix. However, while the eigenvalue decomposition is only applicable to diagonalizable matrices, the Schur decomposition can be applied to any square matrix. Additionally, the diagonal elements in the upper triangular matrices in the Schur decomposition correspond to the eigenvalues of the original matrix, while the columns of the orthogonal matrix in the decomposition correspond to the eigenvectors.

3. What is the significance of the Schur decomposition in linear algebra?

The Schur decomposition is a powerful tool in linear algebra as it allows for a matrix to be represented in a simpler form, making it easier to perform computations and analyze its properties. It is also useful in solving systems of linear equations and in finding the eigenvalues and eigenvectors of a matrix.

4. Can the Schur decomposition be used on non-square matrices?

No, the Schur decomposition is only applicable to square matrices. The resulting upper triangular matrices in the decomposition must have the same dimensions as the original matrix.

5. How is the Schur decomposition calculated?

The Schur decomposition can be calculated using algorithms such as the Schur algorithm or the QR algorithm. These methods involve iteratively applying similarity transformations to the original matrix until it is transformed into upper triangular form. The resulting upper triangular matrices and orthogonal matrix make up the Schur decomposition of the original matrix.

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