8.2E: Orthogonal Diagonalization Exercises

Exercises for 1

solutions

2

Normalize the rows to make each of the following matrices orthogonal.

\(A = \left[ \begin{array}{rr} 1 & 1 \\ -1 & 1 \end{array}\right]\) \(A = \left[ \begin{array}{rr} 3 & -4 \\ 4 & 3 \end{array}\right]\) \(A = \left[ \begin{array}{rr} 1 & 2 \\ -4 & 2 \end{array}\right]\) \(A = \left[ \begin{array}{rr} a & b \\ -b & a \end{array}\right]\), \((a,b) \neq (0,0)\) \(A = \left[ \begin{array}{ccc} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 2 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 2 & 1 & -1 \\ 1 & -1 & 1 \\ 0 & 1 & 1 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 2 & 6 & -3 \\ 3 & 2 & 6 \\ -6 & 3 & 2 \end{array}\right]\)

2. \(\frac{1}{5}\left[ \begin{array}{rr} 3 & -4 \\ 4 & 3 \end{array}\right]\)
3. \(\frac{1}{\sqrt{a^2 + b^2}}\left[ \begin{array}{rr} a & b \\ -b & a \end{array}\right]\)
4. \(\left[ \begin{array}{rrr} \frac{2}{\sqrt{6}} & \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{6}}\\ \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array}\right]\)
5. \(\frac{1}{7}\left[ \begin{array}{rrr} 2 & 6 & -3 \\ 3 & 2 & 6 \\ -6 & 3 & 2 \end{array}\right]\)

If \(P\) is a triangular orthogonal matrix, show that \(P\) is diagonal and that all diagonal entries are \(1\) or \(-1\).

We have \(P^{T} = P^{-1}\); this matrix is lower triangular (left side) and also upper triangular (right side–see Lemma [lem:006547]), and so is diagonal. But then \(P = P^{T} = P^{-1}\), so \(P^{2} = I\). This implies that the diagonal entries of \(P\) are all \(\pm 1\).

If \(P\) is orthogonal, show that \(kP\) is orthogonal if and only if \(k = 1\) or \(k = -1\).

If the first two rows of an orthogonal matrix are \((\frac{1}{3}, \frac{2}{3}, \frac{2}{3})\) and \((\frac{2}{3}, \frac{1}{3}, \frac{-2}{3})\), find all possible third rows.

For each matrix \(A\), find an orthogonal matrix \(P\) such that \(P^{-1}AP\) is diagonal.

\(A = \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right]\) \(A = \left[ \begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 2 & 5 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 3 & 0 & 7 \\ 0 & 5 & 0 \\ 7 & 0 & 3 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right]\) \(A = \left[ \begin{array}{rrr} 5 & -2 & -4 \\ -2 & 8 & -2\\ -4 & -2 & 5 \end{array}\right]\) \(A = \left[ \begin{array}{rrrr} 5 & 3 & 0 & 0 \\ 3 & 5 & 0 & 0 \\ 0 & 0 & 7 & 1 \\ 0 & 0 & 1 & 7 \end{array}\right]\) \(A = \left[ \begin{array}{rrrr} 3 & 5 & -1 & 1 \\ 5 & 3 & 1 & -1 \\ -1 & 1 & 3 & 5 \\ 1 & -1 & 5 & 3 \end{array}\right]\)

2. \(\frac{1}{\sqrt{2}}\left[ \begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right]\)
3. \(\frac{1}{\sqrt{2}}\left[ \begin{array}{rrr} 0 & 1 & 1\\ \sqrt{2} & 0 & 0\\ 0 & 1 & -1 \end{array}\right]\)
4. \(\frac{1}{3\sqrt{2}}\left[ \begin{array}{rrr} 2\sqrt{2} & 3 & 1\\ \sqrt{2} & 0 & -4\\ 2\sqrt{2} & -3 & 1 \end{array}\right]\) or \(\frac{1}{3}\left[ \begin{array}{rrr} 2 & -2 & 1\\ 1 & 2 & 2\\ 2 & 1 & -2 \end{array}\right]\)
5. \(\frac{1}{2}\left[ \begin{array}{rrrr} 1 & -1 & \sqrt{2} & 0\\ -1 & 1 & \sqrt{2} & 0\\ -1 & -1 & 0 & \sqrt{2}\\ 1 & 1 & 0 & \sqrt{2} \end{array}\right]\)

Consider \(A = \left[ \begin{array}{rrr} 0 & a & 0 \\ a & 0 & c \\ 0 & c & 0 \end{array}\right]\) where one of \(a, c \neq 0\). Show that \(c_{A}(x) = x(x - k)(x + k)\), where \(k = \sqrt{a^2 + c^2}\) and find an orthogonal matrix \(P\) such that \(P^{-1}AP\) is diagonal.

\(P = \frac{1}{\sqrt{2}k}\left[ \begin{array}{rrr} c\sqrt{2} & a & a \\ 0 & k & -k \\ -a\sqrt{2} & c & c \end{array}\right]\)

Consider \(A = \left[ \begin{array}{rrr} 0 & 0 & a \\ 0 & b & 0 \\ a & 0 & 0 \end{array}\right]\). Show that \(c_{A}(x) = (x - b)(x - a)(x + a)\) and find an orthogonal matrix \(P\) such that \(P^{-1}AP\) is diagonal.

Given \(A = \left[ \begin{array}{rr} b & a \\ a & b \end{array}\right]\), show that
\(c_{A}(x) = (x - a - b)(x + a - b)\) and find an orthogonal matrix \(P\) such that \(P^{-1}AP\) is diagonal.

Consider \(A = \left[ \begin{array}{rrr} b & 0 & a \\ 0 & b & 0 \\ a & 0 & b \end{array}\right]\). Show that \(c_{A}(x) = (x - b)(x - b - a)(x - b + a)\) and find an orthogonal matrix \(P\) such that \(P^{-1}AP\) is diagonal.

In each case find new variables \(y_{1}\) and \(y_{2}\) that diagonalize the quadratic form \(q\).

\(q = x_{1}^2 + 6x_{1}x_2 + x_{2}^2\) \(q = x_{1}^2 + 4x_{1}x_2 - 2x_{2}^2\)

2. \(y_{1} = \frac{1}{\sqrt{5}}(-x_{1} + 2x_{2})\) and \(y_{2} = \frac{1}{\sqrt{5}}(2x_{1} + x_{2})\); \(q = -3y_{1}^2 + 2y_{2}^2\).

Show that the following are equivalent for a symmetric matrix \(A\).

\(A\) is orthogonal. \(A^{2} = I\). All eigenvalues of \(A\) are \(\pm 1\).

[Hint: For (b) if and only if (c), use Theorem [thm:024303].]

3. \(\Rightarrow\) a. By Theorem [thm:024227] let \(P^{-1}AP = D = \func{diag}(\lambda_{1}, \dots, \lambda_{n})\) where the \(\lambda_{i}\) are the eigenvalues of \(A\). By c. we have \(\lambda_{i} = \pm 1\) for each \(i\), whence \(D^{2} = I\). But then \(A^{2} = (PDP^{-1})^{2} = PD^{2}P^{-1} = I\). Since \(A\) is symmetric this is \(AA^{T} = I\), proving a.

[ex:8_2_12] We call matrices \(A\) and \(B\) orthogonally similar (and write \(A \stackrel{\circ}{\sim} B\)) if \(B = P^{T}AP\) for an orthogonal matrix \(P\).

1. Show that \(A \stackrel{\circ}{\sim} A\) for all \(A\); \(A \stackrel{\circ}{\sim} B \Rightarrow B \stackrel{\circ}{\sim} A\); and \(A \stackrel{\circ}{\sim} B\) and \(B \stackrel{\circ}{\sim} C \Rightarrow A \stackrel{\circ}{\sim} C\).
2. Show that the following are equivalent for two symmetric matrices \(A\) and \(B\).
   1. \(A\) and \(B\) are similar.
   2. \(A\) and \(B\) are orthogonally similar.
   3. \(A\) and \(B\) have the same eigenvalues.

Assume that \(A\) and \(B\) are orthogonally similar (Exercise [ex:8_2_12]).

1. If \(A\) and \(B\) are invertible, show that \(A^{-1}\) and \(B^{-1}\) are orthogonally similar.
2. Show that \(A^{2}\) and \(B^{2}\) are orthogonally similar.
3. Show that, if \(A\) is symmetric, so is \(B\).

2. If \(B = P^{T}AP = P^{-1}\), then \(B^{2} = P^{T}APP^{T}AP = P^{T}A^{2}P\).

If \(A\) is symmetric, show that every eigenvalue of \(A\) is nonnegative if and only if \(A = B^{2}\) for some symmetric matrix \(B\).

[ex:8_2_15] Prove the converse of Theorem [thm:024396]:

If \((A\mathbf{x})\bullet \mathbf{y} = \mathbf{x}\bullet (A\mathbf{y})\) for all \(n\)-columns \(\mathbf{x}\) and \(\mathbf{y}\), then \(A\) is symmetric.

If \(\mathbf{x}\) and \(\mathbf{y}\) are respectively columns \(i\) and \(j\) of \(I_{n}\), then \(\mathbf{x}^{T}A^{T}\mathbf{y} = \mathbf{x}^{T}A\mathbf{y}\) shows that the \((i, j)\)-entries of \(A^{T}\) and \(A\) are equal.

Show that every eigenvalue of \(A\) is zero if and only if \(A\) is nilpotent (\(A^{k} = 0\) for some \(k \geq 1\)).

If \(A\) has real eigenvalues, show that \(A = B + C\) where \(B\) is symmetric and \(C\) is nilpotent.

Let \(P\) be an orthogonal matrix.

1. Show that \(\det P = 1\) or \(\det P = -1\).
2. Give \(2 \times 2\) examples of \(P\) such that \(\det P = 1\) and \(\det P = -1\).
3. If \(\det P = -1\), show that \(I + P\) has no inverse. [Hint: \(P^{T}(I + P) = (I + P)^{T}\).]
4. [Hint: \(P^{T}(I - P) = -(I - P)^{T}\).]

2. \(\det \left[ \begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right] = 1\)
   and \(\det \left[ \begin{array}{rr} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{array}\right] = -1\)

   [Remark: These are the only \(2 \times 2\) examples.]

3. Use the fact that \(P^{-1} = P^{T}\) to show that \(P^{T}(I - P) = -(I - P)^{T}\). Now take determinants and use the hypothesis that \(\det P \neq (-1)^{n}\).

We call a square matrix \(E\) a projection matrix if \(E^{2} = E = E^{T}\). (See Exercise [ex:8_1_17].)

1. If \(E\) is a projection matrix, show that \(P = I - 2E\) is orthogonal and symmetric.

2. If \(P\) is orthogonal and symmetric, show that
   \(E = \frac{1}{2}(I - P)\) is a projection matrix.

3. If \(U\) is \(m \times n\) and \(U^{T}U = I\) (for example, a unit column in \(\mathbb{R}^n\)), show that \(E = UU^{T}\) is a projection matrix.

A matrix that we obtain from the identity matrix by writing its rows in a different order is called a permutation matrix. Show that every permutation matrix is orthogonal.

If the rows \(\mathbf{r}_{1}, \dots, \mathbf{r}_{n}\) of the \(n \times n\) matrix \(A = \left[ a_{ij} \right]\) are orthogonal, show that the \((i, j)\)-entry of \(A^{-1}\) is \(\frac{a_{ji}}{\| \mathbf{r}_{j} \|^2}\).

We have \(AA^{T} = D\), where \(D\) is diagonal with main diagonal entries \(\| R_{1}\|^{2}, \dots, \| R_{n}\|^{2}\). Hence \(A^{-1} = A^{T}D^{-1}\), and the result follows because \(D^{-1}\) has diagonal entries \(1 / \| R_{1}\|^{2}, \dots, 1 / \| R_{n}\|^{2}\).

1. Let \(A\) be an \(m \times n\) matrix. Show that the following are equivalent.
   1. \(A\) has orthogonal rows.
   2. \(A\) can be factored as \(A = DP\), where \(D\) is invertible and diagonal and \(P\) has orthonormal rows.
   3. \(AA^{T}\) is an invertible, diagonal matrix.
2. Show that an \(n \times n\) matrix \(A\) has orthogonal rows if and only if \(A\) can be factored as \(A = DP\), where \(P\) is orthogonal and \(D\) is diagonal and invertible.

Let \(A\) be a skew-symmetric matrix; that is, \(A^{T} = -A\). Assume that \(A\) is an \(n \times n\) matrix.

1. Show that \(I + A\) is invertible. [Hint: By Theorem [thm:004553], it suffices to show that \((I + A)\mathbf{x} = \mathbf{0}\), \(\mathbf{x}\) in \(\mathbb{R}^n\), implies \(\mathbf{x} = \mathbf{0}\). Compute \(\mathbf{x}\bullet \mathbf{x} = \mathbf{x}^{T}\mathbf{x}\), and use the fact that \(A\mathbf{x} = -\mathbf{x}\) and \(A^{2}\mathbf{x} = \mathbf{x}\).]
2. Show that \(P = (I - A)(I + A)^{-1}\) is orthogonal.
3. Show that every orthogonal matrix \(P\) such that \(I + P\) is invertible arises as in part (b) from some skew-symmetric matrix \(A\).

2. Because \(I - A\) and \(I + A\) commute, \(PP^{T} = (I - A)(I + A)^{-1}[(I + A)^{-1}]^{T}(I - A)^{T} = (I - A)(I + A)^{-1}(I - A)^{-1}(I + A) = I\).

Show that the following are equivalent for an \(n \times n\) matrix \(P\).

1. \(P\) is orthogonal.
2. \(\| P\mathbf{x}\| = \|\mathbf{x}\|\) for all columns \(\mathbf{x}\) in \(\mathbb{R}^n\).
3. \(\| P\mathbf{x} - P\mathbf{y}\| = \|\mathbf{x} - \mathbf{y}\|\) for all columns \(\mathbf{x}\) and \(\mathbf{y}\) in \(\mathbb{R}^n\).
4. [Hints: For (c) \(\Rightarrow\) (d), see Exercise [ex:5_3_14](a). For (d) \(\Rightarrow\) (a), show that column \(i\) of \(P\) equals \(P\mathbf{e}_{i}\), where \(\mathbf{e}_{i}\) is column \(i\) of the identity matrix.]

Show that every \(2 \times 2\) orthogonal matrix has the form \(\left[ \begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right]\) or \(\left[ \begin{array}{rr} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{array}\right]\) for some angle \(\theta\).

Use Theorem [thm:024503] to show that every symmetric matrix is orthogonally diagonalizable.