Let A be a n × n matrix with Schur decomposition UTUH. Show that if the diagonal entries of T are all distinct, then there is an upper triangular matrix R such that X = UR diagonalizes A.
Let A be a n × n matrix with Schur decomposition
UTUH. Show that if the diagonal entries of T are
all distinct, then there is an upper triangular matrix
R such that X = UR diagonalizes A.
The Schur decomposition of a square matrix A states that there exists an orthogonal matrix U and an upper triangular matrix T such that A = UTU^H, where H denotes the Hermitian transpose (i.e., the conjugate transpose).
Suppose that the diagonal entries of T are all distinct. Our goal is to find an upper triangular matrix R such that X = UR diagonalizes A, meaning that X^{-1}AX is a diagonal matrix.
To do so, we can follow these steps:
Note that A = UTU^H implies that A^H = (UTU^H)^H = UT^HU^H, since U is orthogonal.
Compute the matrix product T^HUT. Since U is orthogonal, we have
T^HUT = T^{-1}(UT)T = T^{-1}AT
by left-multiplying A = UTU^H with T^{-1}.
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