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For a rectangular matrix A, how are the eigenvalues of ATA and AAT related?

For a rectangular matrix A, how are the eigenvalues of ATA and AAT related?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Transcription:** "For a rectangular matrix \( A \), how are the eigenvalues of \( A^T A \) and \( A A^T \) related?" **Explanation:** This image contains a question centered on linear algebra and the relationship between eigenvalues. It asks about the connection between the eigenvalues of the matrices \( A^T A \) and \( A A^T \), where \( A \) is a rectangular matrix. \( A^T \) represents the transpose of matrix \( A \). In terms of understanding the concepts: - **Rectangular Matrix \( A \)**: A matrix with a different number of rows and columns. - **Eigenvalues**: Scalars that provide insight into the intrinsic properties of matrices in linear transformations. - **\( A^T A \) and \( A A^T \)**: These are products of the matrices that often appear in the context of Singular Value Decomposition (SVD). The key discussion point is how these products relate to the original matrix's properties, notably in applications like Principal Component Analysis (PCA) and other dimensionality reduction techniques.
Transcribed Image Text:**Transcription:** "For a rectangular matrix \( A \), how are the eigenvalues of \( A^T A \) and \( A A^T \) related?" **Explanation:** This image contains a question centered on linear algebra and the relationship between eigenvalues. It asks about the connection between the eigenvalues of the matrices \( A^T A \) and \( A A^T \), where \( A \) is a rectangular matrix. \( A^T \) represents the transpose of matrix \( A \). In terms of understanding the concepts: - **Rectangular Matrix \( A \)**: A matrix with a different number of rows and columns. - **Eigenvalues**: Scalars that provide insight into the intrinsic properties of matrices in linear transformations. - **\( A^T A \) and \( A A^T \)**: These are products of the matrices that often appear in the context of Singular Value Decomposition (SVD). The key discussion point is how these products relate to the original matrix's properties, notably in applications like Principal Component Analysis (PCA) and other dimensionality reduction techniques.
**Transcription for Educational Website:** **Title: Eigenvalues of Transposed Matrices** **For a rectangular matrix \( A \), how are the eigenvalues of \( A^T A \) and \( A A^T \) related?** **Explanation:** In the context of linear algebra and matrix theory, the matrices \( A^T A \) and \( A A^T \) are both important constructions derived from a given rectangular matrix \( A \). These matrices are known as the Gramian matrices or Gram matrices. 1. **Matrix \( A \):** - \( A \) is a rectangular matrix, meaning its number of rows is not necessarily equal to its number of columns. This makes \( A \) either a tall or a wide matrix, depending on its dimensions. 2. **Matrix \( A^T A \):** - The product \( A^T A \) is always a square matrix, regardless of the original dimensions of \( A \). - It is of size \( n \times n \) if \( A \) is \( m \times n \). 3. **Matrix \( A A^T \):** - Similarly, \( A A^T \) is also a square matrix. - It is of size \( m \times m \) if \( A \) is \( m \times n \). **Eigenvalues Relationship:** The eigenvalues of \( A^T A \) and \( A A^T \) have a special relationship: - Both \( A^T A \) and \( A A^T \) have non-negative eigenvalues. - The non-zero eigenvalues of \( A^T A \) are the same as the non-zero eigenvalues of \( A A^T \). This is due to the fact that they share the same singular values, which are the square roots of the non-zero eigenvalues. - The number of non-zero eigenvalues is equal to the rank of \( A \). Understanding the relationship between these matrices and their eigenvalues is fundamental in applications such as Principal Component Analysis (PCA), where these concepts are used to find important directions in data.
Transcribed Image Text:**Transcription for Educational Website:** **Title: Eigenvalues of Transposed Matrices** **For a rectangular matrix \( A \), how are the eigenvalues of \( A^T A \) and \( A A^T \) related?** **Explanation:** In the context of linear algebra and matrix theory, the matrices \( A^T A \) and \( A A^T \) are both important constructions derived from a given rectangular matrix \( A \). These matrices are known as the Gramian matrices or Gram matrices. 1. **Matrix \( A \):** - \( A \) is a rectangular matrix, meaning its number of rows is not necessarily equal to its number of columns. This makes \( A \) either a tall or a wide matrix, depending on its dimensions. 2. **Matrix \( A^T A \):** - The product \( A^T A \) is always a square matrix, regardless of the original dimensions of \( A \). - It is of size \( n \times n \) if \( A \) is \( m \times n \). 3. **Matrix \( A A^T \):** - Similarly, \( A A^T \) is also a square matrix. - It is of size \( m \times m \) if \( A \) is \( m \times n \). **Eigenvalues Relationship:** The eigenvalues of \( A^T A \) and \( A A^T \) have a special relationship: - Both \( A^T A \) and \( A A^T \) have non-negative eigenvalues. - The non-zero eigenvalues of \( A^T A \) are the same as the non-zero eigenvalues of \( A A^T \). This is due to the fact that they share the same singular values, which are the square roots of the non-zero eigenvalues. - The number of non-zero eigenvalues is equal to the rank of \( A \). Understanding the relationship between these matrices and their eigenvalues is fundamental in applications such as Principal Component Analysis (PCA), where these concepts are used to find important directions in data.
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