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東京大学 新領域創成科学研究科 メディカル情報生命専攻 2024年1月実施 問題8

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zephyr

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Suppose that the eigenvalues and the corresponding eigenvectors of an square matrix are and respectively.

Suppose that is the identity matrix, and the inverse matrix of an invertible matrix is .

Answer the following questions.

  1. Show all the eigenvalues and the corresponding eigenvectors of .

  2. If are mutually different, show that is a diagonal matrix, using that is a matrix of concatenated eigenvectors.

  3. Show all the eigenvalues and the corresponding eigenvectors of .

Misplaced & 3 & 0 & 0 \\ -2 & 3 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$ 4. Suppose that $\mu$ is the maximum eigenvalue of $\mathbf{B}$, and $\mathbf{\gamma} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$. Calculate $\mathbf{\delta} = (\mathbf{B} - \mu \mathbf{I}_3)\mathbf{\gamma}$. 5. Suppose that $\beta$ is the eigenvector of $\mathbf{B}$ corresponding to the minimum eigenvalue. Calculate $\mathbf{Q}^{-1}\mathbf{B}\mathbf{Q}$ using $\mathbf{Q} = (\mathbf{\delta}, \mathbf{\gamma}, \beta)$ that is a matrix concatenating $\mathbf{\delta}, \mathbf{\gamma}, \beta$. 6. Suppose that $m$ is an arbitrary positive integer. Calculate $\mathbf{B}^m$. --- 假设 $n \times n$ 方阵 $\mathbf{A}$ 的特征值及相应的特征向量分别为 $\lambda_1, \dots, \lambda_n$ 和 $\mathbf{\alpha}_1, \dots, \mathbf{\alpha}_n$。 假设 $\mathbf{I}_n$ 是 $n \times n$ 的单位矩阵，并且可逆矩阵 $\mathbf{C}$ 的逆矩阵为 $\mathbf{C}^{-1}$。 回答以下问题。 1. 展示 $\mathbf{A}^2$ 的所有特征值及相应的特征向量。 2. 如果 $\lambda_1, \dots, \lambda_n$ 是互不相同的，证明 $\mathbf{P}^{-1} \mathbf{A} \mathbf{P}$ 是一个对角矩阵，其中 $\mathbf{P} = (\mathbf{\alpha}_1, \dots, \mathbf{\alpha}_n)$ 是由特征向量构成的矩阵。 3. 展示 $\mathbf{B}$ 的所有特征值及相应的特征向量。 $$\mathbf{B} = \begin{pmatrix} 3 & 0 & 0 \\ -2 & 3 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$ 4. 假设 $\mu$ 是 $\mathbf{B}$ 的最大特征值，并且 $\mathbf{\gamma} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$。 计算 $\mathbf{\delta} = (\mathbf{B} - \mu \mathbf{I}_3)\mathbf{\gamma}$。 5. 假设 $\beta$ 是 $\mathbf{B}$ 对应于最小特征值的特征向量。计算 $\mathbf{Q}^{-1}\mathbf{B}\mathbf{Q}$，其中 $\mathbf{Q} = (\mathbf{\delta}, \mathbf{\gamma}, \beta)$ 是由 $\mathbf{\delta}, \mathbf{\gamma}, \beta$ 构成的矩阵。 6. 假设 $m$ 是任意正整数。计算 $\mathbf{B}^m$。 ## **Kai** ### 1. Positive Eigenvalues and Normalized Eigenvectors of $\mathbf{A}^T \mathbf{A}$ Given the singular value decomposition (SVD) of $\mathbf{A}$ as $\mathbf{A} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T$, we can express $\mathbf{A}^T \mathbf{A}$ as follows:

\mathbf{A}^T \mathbf{A} = (\mathbf{U} \mathbf{\Sigma} \mathbf{V}^T)^T (\mathbf{U} \mathbf{\Sigma} \mathbf{V}^T) = \mathbf{V} \mathbf{\Sigma}^T \mathbf{U}^T \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T = \mathbf{V} \mathbf{\Sigma}^2 \mathbf{V}^T

You can't use 'macro parameter character #' in math mode The matrix $\mathbf{\Sigma}^2$ is diagonal with the diagonal elements $\sigma_k^2$ ($k = 1, \ldots, r$). Thus, the positive eigenvalues of $\mathbf{A}^T \mathbf{A}$ are exactly the $\sigma_k^2$, and the associated normalized eigenvectors are the columns of $\mathbf{V}$. ### 2. Surjectivity and Injectivity of $T_{\mathbf{A}}$ **Surjective (onto)**: The mapping $T_{\mathbf{A}}: \mathbb{R}^m \to \mathbb{R}^n$ is surjective if the range of $\mathbf{A}$ spans $\mathbb{R}^n$, i.e., $\mathbf{A}$ has full row rank. This occurs when $r = n \leq m$. **Injective (one-to-one)**: The mapping $T_{\mathbf{A}}$ is injective if the kernel of $\mathbf{A}$ contains only the zero vector, i.e., $\mathbf{A}$ has full column rank. This occurs when $r = m \leq n$. ### 3. Image of $T_{\mathbf{B}}$ and Kernel of $T_{\mathbf{A}}$ The pseudoinverse $\mathbf{A}^+$ is defined as $\mathbf{A}^+ = \mathbf{V} \mathbf{\Sigma}^{-1} \mathbf{U}^T$. Consider $\mathbf{B} = \mathbf{I}_m - \mathbf{A}^+ \mathbf{A}$. We need to show that $\mathrm{Im}(T_{\mathbf{B}})$ is isomorphic to $\mathrm{Ker}(T_{\mathbf{A}})$. Observe the following:

\mathbf{B} \mathbf{A} = (\mathbf{I}_m - \mathbf{A}^+ \mathbf{A}) \mathbf{A} = \mathbf{A} - \mathbf{A}^+ \mathbf{A} \mathbf{A} = \mathbf{A} - \mathbf{A} = \mathbf{0}

\mathbf{B} \mathbf{x} = (\mathbf{I}_m - \mathbf{A}^+ \mathbf{A}) \mathbf{x} = \mathbf{x}

You can't use 'macro parameter character #' in math mode Thus, $\mathbf{x} \in \mathrm{Im}(\mathbf{B})$. Therefore, $\mathrm{Im}(\mathbf{B}) = \mathrm{Ker}(\mathbf{A})$. ### 4. Orthogonal Decomposition Given $\mathbf{x} = \mathbf{x}_1 + \mathbf{x}_2$ where $\mathbf{x}_1 = \mathbf{B} \mathbf{x}$ and $\mathbf{x}_2 = \mathbf{x} - \mathbf{x}_1$:

\mathbf{x}_2 = \mathbf{x} - \mathbf{B} \mathbf{x} = \mathbf{x} - (\mathbf{I}_m - \mathbf{A}^+ \mathbf{A}) \mathbf{x} = \mathbf{A}^+ \mathbf{A} \mathbf{x}

\mathbf{x}_1^T \mathbf{x}_2 = (\mathbf{B} \mathbf{x})^T (\mathbf{A}^+ \mathbf{A} \mathbf{x}) = \mathbf{x}^T \mathbf{B}^T \mathbf{A}^+ \mathbf{A} \mathbf{x}

\mathbf{x}^T (\mathbf{I}_m - \mathbf{A}^+ \mathbf{A}) \mathbf{A}^+ \mathbf{A} \mathbf{x} = \mathbf{x}^T (\mathbf{A}^+ \mathbf{A} - \mathbf{A}^+ \mathbf{A}) \mathbf{x} = \mathbf{0}

You can't use 'macro parameter character #' in math mode Thus, $\mathbf{x}_1$ and $\mathbf{x}_2$ are orthogonal. ### 5. Minimizing $(\mathbf{A} \mathbf{x} - \mathbf{b})^T (\mathbf{A} \mathbf{x} - \mathbf{b})$ Let $\mathbf{x}_0 = \mathbf{A}^+ \mathbf{b}$. We need to show that $\mathbf{x} = \mathbf{x}_0$ minimizes the expression. Consider the error:

\mathbf{A} \mathbf{x} - \mathbf{b} = \mathbf{A} (\mathbf{x} - \mathbf{x}_0) + (\mathbf{A} \mathbf{x}_0 - \mathbf{b})

\mathbf{A} \mathbf{x} - \mathbf{b} = \mathbf{A} (\mathbf{x} - \mathbf{x}_0)

(\mathbf{A} \mathbf{x} - \mathbf{b})^T (\mathbf{A} \mathbf{x} - \mathbf{b}) = (\mathbf{A} (\mathbf{x} - \mathbf{x}_0))^T (\mathbf{A} (\mathbf{x} - \mathbf{x}_0))

You can't use 'macro parameter character #' in math mode This is minimized when $\mathbf{x} = \mathbf{x}_0$ since $\mathbf{A} \mathbf{x}_0 = \mathbf{b}$ and $\mathbf{A} (\mathbf{x} - \mathbf{x}_0) = \mathbf{0}$. ## **Knowledge** ### 重点词汇 - singular value decomposition (SVD) 奇异值分解 - pseudoinverse 广义逆 - surjective 满射 - injective 单射 - orthogonal decomposition 正交分解 ### 参考资料 1. "Linear Algebra and Its Applications" by Gilbert Strang, Chapter 7: The Singular Value Decomposition (SVD) 2. "Matrix Computations" by Gene H. Golub and Charles F. Van Loan, Chapter 2: Matrix Analysis