Subsection 26.3

Orthogonalization (26.42-26.48)

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Problem 26.42

Orthogonalize the following systems of vectors of an arithmetic space with the standard scalar product:

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(1)

[13−2]\left[\begin{smallmatrix} 1 \\ 3 \\ -2\end{smallmatrix}\right], [37−2]\left[\begin{smallmatrix} 3 \\ 7 \\ -2\end{smallmatrix}\right];

(2)

[210−1],[3626]\left[\begin{smallmatrix} 2 \\ 1 \\ 0 \\ - 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 3 \\ 6 \\ 2 \\ 6\end{smallmatrix}\right]

(3)

[131],[513],[16−8]\left[\begin{smallmatrix} 1 \\ 3 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 5 \\ 1 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 1 \\ 6 \\ - 8\end{smallmatrix}\right]

(4)

[212],[622],[14−3]\left[\begin{smallmatrix} 2 \\ 1 \\ 2\end{smallmatrix}\right],\left[\begin{smallmatrix} 6 \\ 2 \\ 2\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 4 \\ - 3\end{smallmatrix}\right]

(5)

[123],[211],[6−7−2]\left[\begin{smallmatrix} 1 \\ 2 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 \\ 1 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 6 \\ -7 \\ - 2\end{smallmatrix}\right]

(6)

[1212]\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 2\end{smallmatrix}\right], [4041]\left[\begin{smallmatrix} 4 \\ 0 \\ 4 \\ 1\end{smallmatrix}\right], [113−1−3]\left[\begin{smallmatrix} 1 \\ 13 \\ -1 \\ -3\end{smallmatrix}\right]

(7)

[1−1−11]\left[\begin{smallmatrix} 1 \\ - 1 \\ - 1 \\ 1\end{smallmatrix}\right], [2332]\left[\begin{smallmatrix} 2 \\ 3 \\ 3 \\ 2\end{smallmatrix}\right], [4402]\left[\begin{smallmatrix} 4 \\ 4 \\ 0 \\ 2\end{smallmatrix}\right], [1−5−5−1]\left[\begin{smallmatrix} 1 \\ -5 \\ -5 \\ -1\end{smallmatrix}\right]

Problem 26.43

In a Euclidean space E\mathcal{E} an orthonormal basis e is chosen. In it are given the coordinates of the vectors of a basis a of the subspace L⊆E\mathcal{L} \subseteq \mathcal{E}. Using the orthogonalization process, find in ee the coordinate columns of the vectors of an orthonormal basis in L\mathcal{L} and write out the transition matrix from the basis a to this basis:

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(1)

[31],[9−7]\left[\begin{smallmatrix} 3 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 9 \\ -7\end{smallmatrix}\right]

(2)

[110],[20−1],[003]\left[\begin{smallmatrix} 1 \\ 1 \\ 0\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 \\ 0 \\ - 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 0 \\ 0 \\ 3\end{smallmatrix}\right]

(3)

[11−22]\left[\begin{smallmatrix} 1 \\ 1 \\ -2\sqrt{2}\end{smallmatrix}\right], [3−1−2]\left[\begin{smallmatrix} 3 \\ -1 \\ -\sqrt{2}\end{smallmatrix}\right], [42−2]\left[\begin{smallmatrix} 4 \\ 2 \\ -\sqrt{2}\end{smallmatrix}\right]

(4)

[111−1]\left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ - 1\end{smallmatrix}\right], [331−1]\left[\begin{smallmatrix} 3 \\ 3 \\ 1 \\ - 1\end{smallmatrix}\right], [31−13]\left[\begin{smallmatrix} 3 \\ 1 \\ - 1 \\ 3\end{smallmatrix}\right], [1−133]\left[\begin{smallmatrix} 1 \\ - 1 \\ 3 \\ 3\end{smallmatrix}\right]

(5)

[1212]\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 2\end{smallmatrix}\right], [2301]\left[\begin{smallmatrix} 2 \\ 3 \\ 0 \\ 1\end{smallmatrix}\right], [32−12]\left[\begin{smallmatrix} 3 \\ 2 \\ -1 \\ 2\end{smallmatrix}\right], [4121]\left[\begin{smallmatrix} 4 \\ 1 \\ 2 \\ 1\end{smallmatrix}\right]

(6)

[1423]\left[\begin{smallmatrix} 1 \\ 4 \\ 2 \\ 3\end{smallmatrix}\right], [1503]\left[\begin{smallmatrix} 1 \\ 5 \\ 0 \\ 3\end{smallmatrix}\right], [−1927]\left[\begin{smallmatrix} -1 \\ 9 \\ 2 \\ 7\end{smallmatrix}\right]

(7)

[4−2−10],[9−2−20],[−3−1111]\left[\begin{smallmatrix} 4 \\ -2 \\ -1 \\ 0\end{smallmatrix}\right],\left[\begin{smallmatrix} 9 \\ -2 \\ -2 \\ 0\end{smallmatrix}\right],\left[\begin{smallmatrix} -3 \\ -1 \\ 11 \\ 1\end{smallmatrix}\right]

(8)

[1213],[4326],[43−74]\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 4 \\ 3 \\ 2 \\ 6\end{smallmatrix}\right],\left[\begin{smallmatrix} 4 \\ 3 \\ -7 \\ 4\end{smallmatrix}\right]

Problem 26.44

Orthogonalize and normalize the system of vectors given in a basis ee by their coordinate columns. The Gram matrix Γ\Gamma of the basis ee is given:

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(1)

[13],[24],Γ=A19\left[\begin{smallmatrix} 1 \\ 3\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 \\ 4\end{smallmatrix}\right], \quad \Gamma = \mathbf{A}_{19}. A19=[1112]\mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right].

(2)

[120]\left[\begin{smallmatrix} 1 \\ 2 \\ 0\end{smallmatrix}\right], [203]\left[\begin{smallmatrix} 2 \\ 0 \\ 3\end{smallmatrix}\right], [186]\left[\begin{smallmatrix} 1 \\ 8 \\ 6\end{smallmatrix}\right], Γ=A387\Gamma = \mathbf{A}_{387}. A387=[210121012]\mathbf{A}_{387} = \left[\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{smallmatrix}\right].

(3)

[111]\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], [421]\left[\begin{smallmatrix} 4 \\ 2 \\ 1\end{smallmatrix}\right], [19−8]\left[\begin{smallmatrix} 1 \\ 9 \\ - 8\end{smallmatrix}\right], Γ=A386\Gamma = \mathbf{A}_{386}. A386=[20−1021−112]\mathbf{A}_{386} = \left[\begin{smallmatrix} 2 & 0 & -1 \\ 0 & 2 & 1 \\ -1 & 1 & 2 \end{smallmatrix}\right].

(4)

[−321]\left[\begin{smallmatrix} - 3 \\ 2 \\ 1\end{smallmatrix}\right], [−854]\left[\begin{smallmatrix} - 8 \\ 5 \\ 4\end{smallmatrix}\right], [240]\left[\begin{smallmatrix} 2 \\ 4 \\ 0\end{smallmatrix}\right], Γ=A207\Gamma = \mathbf{A}_{207}. A207=[12025−20−25]\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right].

Problem 26.45
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(1)

Using the orthogonalization process, prove that a nondegenerate square matrix can be decomposed into the product of an orthogonal matrix QQ and an upper triangular matrix RR with positive diagonal elements (QR decomposition).

(2)

Prove the uniqueness of the QRQR decomposition of a nondegenerate matrix.

Problem 26.46

Find the QRQR decomposition of the given matrix (problem 26. 45):

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(1)

[1−216]\left[\begin{smallmatrix} 1 & - 2 \\ 1 & 6\end{smallmatrix}\right]

(2)

[391−7]\left[\begin{smallmatrix} 3 & 9 \\ 1 & - 7\end{smallmatrix}\right]

(3)

[12010001−1]\left[\begin{smallmatrix} 1 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & - 1\end{smallmatrix}\right]

(4)

[1341−12−22−22−2]\left[\begin{smallmatrix} 1 & 3 & 4 \\ 1 & - 1 & 2 \\ - 2\sqrt{2} & - 2\sqrt{2} & - \sqrt{2}\end{smallmatrix}\right]

(5)

[1331131−111−13−1−133]\left[\begin{smallmatrix} 1 & 3 & 3 & 1 \\ 1 & 3 & 1 & - 1 \\ 1 & 1 & - 1 & 3 \\ - 1 & - 1 & 3 & 3\end{smallmatrix}\right]

(6)

[1234232110−122121]\left[\begin{smallmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 2 & 1 \\ 1 & 0 & - 1 & 2 \\ 2 & 1 & 2 & 1\end{smallmatrix}\right].

Problem 26.47

Prove that the Gram matrix of an arbitrary basis can be decomposed into the product Γ=RTR\Gamma = R^{T} R, where RR is an upper triangular matrix with positive elements on the diagonal. (Compare this with problem 25. 29.)

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Problem 26.48

In a Euclidean space the Gram matrix of a basis f\mathbf{f} is known. Find the transition matrix to the orthonormal basis obtained by orthogonalizing f\mathbf{f}:

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(1)

Γ=[25513]\Gamma =\left[\begin{smallmatrix} 2 & 5 \\ 5 & 13\end{smallmatrix}\right]

(2)

Γ=[13313]\Gamma =\left[\begin{smallmatrix} 1 & 3 \\ 3 & 13\end{smallmatrix}\right]

(3)

Γ=[1232583814]\Gamma =\left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 5 & 8 \\ 3 & 8 & 14\end{smallmatrix}\right]

(4)

Γ=[40040440048040020]\Gamma =\left[\begin{smallmatrix} 4 & 0 & 0 & 4 \\ 0 & 4 & 4 & 0 \\ 0 & 4 & 8 & 0 \\ 4 & 0 & 0 & 20\end{smallmatrix}\right].