Questions and Answers
What is the representation of graphs as matrices?
What is a stochastic matrix?
A continuous graph that preserves its weight under flow.
What does Av = λv represent?
A matrix is positive definite if all of its eigenvalues are less than 0.
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What is the trace of a matrix?
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What is deflation in the context of linear maps?
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What components make up the singular value decomposition of a matrix?
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Data whitening normalizes datasets to have a mean of ______ and a covariance of ______.
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Study Notes
Graphs Represented as Matrices
- Graphs can be represented using an NxN matrix for N nodes.
- Matrix element i,j is 1 if an edge exists from node i to j; otherwise, it is 0.
- Edges may have weights, with values for i,j being greater than or equal to 1.
- In-degree is calculated by summing each column; out-degree by summing each row.
Discrete-Continuous Interchange
- Discrete time points can be represented with a vector of length N for each graph node.
- The vector's value at index n indicates the current weight of that node.
- A normalized NxN graph matrix represents a continuous graph.
- Multiplying the matrix with a discrete point yields a vector for the subsequent time point.
- If total out-degree sums to 1, the graph maintains weight under flow, termed a stochastic matrix.
Eigenvectors, Eigenvalues, and Eigenspectrums
- An eigenvector maintains its homogeneity under a linear map.
- The eigenvalue (λ) is the scalar that scales the eigenvector (v) during transformation: (Av = λv).
- The eigenspectrum consists of all eigenvalues sorted in descending order of magnitude.
Leading Eigenvector and Deflation of Linear Maps
- The leading eigenvector can be determined using a power iteration method.
- Subsequent eigenvectors are found by removing the previous eigenvector from the linear map and applying power iteration again, a process known as deflation.
Principal Component Analysis
- Involves computing the eigenspectrum of the covariance matrix.
- Reveals the most significant axis impacted by the linear transformation.
Trace and Determinant
- The trace of a matrix is the sum of its diagonal elements or the sum of its eigenvalues.
- The determinant is the product of a matrix's eigenvalues.
Definiteness
- A matrix is positive definite if all eigenvalues are greater than zero.
- It is negative definite if all eigenvalues are less than zero.
- Semi-positive/negative matrices have eigenvalues that are either greater or less than or equal to zero.
Singular Value Decomposition
- A matrix can be decomposed into three components represented as:
- ( A[m,n] = U[m,m] \cdot D[m,n] \cdot V[n,n] )
- U is an orthogonal matrix with dimensions m x r, composed of eigenvectors of ( AA^T ).
- D is a diagonal matrix with values equal to the square roots of the eigenvalues of ( A^TA ).
- V is the r x n matrix comprised of eigenvectors of ( A^TA ).
Data Whitening
- A normalization preprocessing technique for datasets.
- Ensures the matrix has a mean of 0 and a covariance of -1.
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Explore the key concepts of computational linear algebra through our flashcards. Understand how graphs are represented as matrices and the importance of discrete-continuous interchanges. Perfect for students looking to deepen their grasp of linear algebra applications.
