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Chapter 6: Problem 20

Let \(A\) be a \(n \times n\) matrix with Schur decomposition \(U T U^{H} .\) Show that if the diagonal entries of \(T\) are all distinct, then there is an upper triangular matrix \(R\) such that \(X=U R\) diagonalizes \(A\)

Short Answer

Expert verified
In conclusion, we have found that if A is a given matrix with a known Schur decomposition that has distinct diagonal entries, then A can be diagonalized using an upper-triangular matrix R (equal to T) and a diagonal matrix D. The diagonal matrix D can be found using the equation \( D = U^H A U \) and the required upper-triangular matrix R is equal to T. Therefore, the matrix X = UR diagonalizes A with A = XDX^(-1).

Step by step solution

01

Rewrite the Given Schur Decomposition

Recall the Schur decomposition of A: \[ A = UTU^H \]
02

Express A as a Diagonal Matrix

We want to find a diagonal matrix D and an upper-triangular matrix R such that A = XDX^(-1), where X = UR. We know that A = UTU^H, so let's find a matrix \(U^H R^{鈭�1} U\) that makes UTU^H diagonal. Rearrange the Schur decomposition equation to resemble the diagonalization form: \[ D = U^H A U \] Since T is an upper-triangular matrix with distinct diagonal entries and U is a unitary matrix, we can use this equation to find D.
03

Find Matrix R

Now, we have X = UR and A = XDX^(-1). We can rewrite this as A = (UR)D(UR)^(-1). Let's find R by comparing this equation to the Schur decomposition (A = UTU^H). From A = (UR)D(UR)^(-1), note that U is the same in both equations if R is chosen to be equal to T (since U has to be unitary): \[ R = T \] Now we have X = UT.
04

Express Matrix A in Terms of Matrix X

Since X = UT and A = UTU^H, we can plug in X to express A in terms of X: \[ A = XU^H \]
05

Confirm A is Diagonalized

Given A = XDX^(-1), we have done the following: 1. Found the diagonal matrix D = U^H A U. 2. Found the upper-triangular matrix R = T. 3. Shown that A = XU^H, where X = UR. We've shown that A can be diagonalized with a diagonal matrix D and an upper-triangular matrix R. Therefore, there is an upper-triangular matrix R (equal to T) such that X = UR diagonalizes A.

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Most popular questions from this chapter

Show that \(A\) and \(A^{T}\) have the same nonzero singular values. How are their singular value decompositions related?

Given that \\[ A=\left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 1 & i \\ 0 & -i & 1 \end{array}\right) \\] find a matrix \(B\) such that \(B^{H} B=A\)

For each of the following, find a matrix \(B\) such that \(B^{2}=A\) (a) \(A=\left(\begin{array}{rr}2 & 1 \\ -2 & -1\end{array}\right)\) (b) \(A=\left(\begin{array}{rrr}9 & -5 & 3 \\ 0 & 4 & 3 \\ 0 & 0 & 1\end{array}\right)\)

Let \(A\) be a \(3 \times 3\) symmetric positive definite matrix and suppose that \(\operatorname{det}\left(A_{1}\right)=3, \operatorname{det}\left(A_{2}\right)=6,\) and \(\operatorname{det}\left(A_{3}\right)=8 .\) What would the pivot elements be in the reduction of \(A\) to triangular form, assuming that only row operation III is used in the reduction process?

Let \(T\) be an upper triangular matrix with distinct diagonal entries (i.e., \(t_{i i} \neq t_{j j}\) whenever \(i \neq j\) ). Show that there is an upper triangular matrix \(R\) that diagonalizes \(T\)
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