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Linear Algebra: Eigenvalues & Eigenvectors Video Lecture | Question Bank for GATE Computer Science Engineering - Computer Science Engineering (CSE)
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FAQs on Linear Algebra: Eigenvalues & Eigenvectors Video Lecture - Question Bank for GATE Computer Science Engineering - Computer Science Engineering (CSE)
| 1. What is the significance of eigenvalues and eigenvectors in linear algebra? |  |
Ans. Eigenvalues and eigenvectors play a crucial role in linear algebra. Eigenvalues are scalar values that represent the scaling factor of the eigenvectors. They help us understand how a linear transformation changes the direction of the vectors. Eigenvectors, on the other hand, are non-zero vectors that only change by a scalar factor when a linear transformation is applied. These vectors provide insights into the principal directions of the transformation.
| 2. How can eigenvalues be computed? | |
Ans. To compute eigenvalues, we need to solve the characteristic equation, which is obtained by setting the determinant of the matrix subtracted by a scalar multiple of the identity matrix equal to zero. This equation is represented as |A - λI| = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. By solving this equation, we can find the eigenvalues.
| 3. What are the applications of eigenvalues and eigenvectors in computer science engineering? | |
Ans. Eigenvalues and eigenvectors find various applications in computer science engineering. They are used in image processing for tasks such as image compression, image denoising, and image recognition. In machine learning, eigenvalues and eigenvectors are employed in principal component analysis (PCA) for dimensionality reduction. They are also utilized in graph theory for clustering algorithms and network analysis.
| 4. Can a matrix have more than one eigenvalue? | |
Ans. Yes, a matrix can have multiple eigenvalues. The number of eigenvalues a matrix has is equal to its dimension. However, it is important to note that a matrix may have repeated eigenvalues, known as degenerate eigenvalues. In such cases, the eigenvectors associated with these repeated eigenvalues can be different.
| 5. How can eigenvectors be computed given the eigenvalues? | |
Ans. Once the eigenvalues are determined, we can find the corresponding eigenvectors by solving the system of linear equations (A - λI)v = 0, where A is the matrix, λ is the eigenvalue, I is the identity matrix, and v is the eigenvector. By substituting the computed eigenvalues into this equation and solving it, we can obtain the corresponding eigenvectors.
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Dec 25, 2024 Last updated |
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