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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Suppose that the eigenvalues and the corresponding eigenvectors of an \(n \times n\) square matrix \(\mathbf{A}\) are \(\lambda_1, \dots, \lambda_n\) and \(\mathbf{\alpha}_1, \dots, \mathbf{\alpha}_n\) respectively.
Suppose that \(\mathbf{I}_n\) is the \(n \times n\) identity matrix, and the inverse matrix of an invertible matrix \(\mathbf{C}\) is \(\mathbf{C}^{-1}\).
Answer the following questions.
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Show all the eigenvalues and the corresponding eigenvectors of \(\mathbf{A}^2\).
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If \(\lambda_1, \dots, \lambda_n\) are mutually different, show that \(\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\) is a diagonal matrix, using \(\mathbf{P} = (\mathbf{\alpha}_1, \dots, \mathbf{\alpha}_n)\) that is a matrix of concatenated eigenvectors.
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Show all the eigenvalues and the corresponding eigenvectors of \(\mathbf{B}\).
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Suppose that \(\mu\) is the maximum eigenvalue of \(\mathbf{B}\), and \(\mathbf{\gamma} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\). Calculate \(\mathbf{\delta} = (\mathbf{B} - \mu \mathbf{I}_3)\mathbf{\gamma}\).
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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 \(\mathbf{\delta}, \mathbf{\gamma}, \beta\).
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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。
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展示 \(\mathbf{A}^2\) 的所有特征值及相应的特征向量。
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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。
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展示 \(\mathbf{B}\) 的所有特征值及相应的特征向量。
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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 \(\mathbf{\delta} = (\mathbf{B} - \mu \mathbf{I}_3)\mathbf{\gamma}\)。 -
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。
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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:
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:
Thus, \(\mathrm{Im}(\mathbf{B}) \subseteq \mathrm{Ker}(\mathbf{A})\).
Now, consider \(\mathbf{x} \in \mathrm{Ker}(\mathbf{A})\). Then \(\mathbf{A} \mathbf{x} = \mathbf{0}\), and
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 \(\mathbf{x}_2 = \mathbf{x} - \mathbf{x}_1\):
To show orthogonality:
Since \(\mathbf{B}\) is symmetric (\(\mathbf{B} = \mathbf{I}_m - \mathbf{A}^+ \mathbf{A}\)):
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:
Since \(\mathbf{x}_0 = \mathbf{A}^+ \mathbf{b}\), we have \(\mathbf{A} \mathbf{x}_0 = \mathbf{b}\), thus:
The norm to be minimized is:
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
- "Linear Algebra and Its Applications" by Gilbert Strang, Chapter 7: The Singular Value Decomposition (SVD)
- "Matrix Computations" by Gene H. Golub and Charles F. Van Loan, Chapter 2: Matrix Analysis