Matrix Algebra and Random Vectors

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Matrix Algebra and Random Vectors

• Shyh-Kang Jeng
• Department of Electrical Engineering/
• Graduate Institute of Communication/
• Graduate Institute of Networking and Multimedia

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General Statistical Distance
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General Statistical Distance
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Quadratic Form
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Statistical Distance under Rotated Coordinate
System
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Rotated Coordinate System
q
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Coordinate Transformation
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Quadratic Form in Transformed Coordinate Systems
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Diagonalized Quadratic Form
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Orthogonal Matrix
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Diagonalization
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Concept of Mapping
x
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Eigenvalues
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Eigenvectors
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Spectral Decomposition
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Positive Definite Matrix

• Matrix A is non-negative definite if
• for all xx1, x2, , xk
• Matrix A is positive definite if
• for all non-zero x

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Positive Definite Matrix
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Inverse and Square-Root Matrix
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Random Vectors and Random Matrices

• Random vector
• Vector whose elements are random variables
• Random matrix
• Matrix whose elements are random variables

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Expected Value of a Random Matrix
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Population Mean Vectors
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Covariance
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Statistically Independent
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Population Variance-Covariance Matrices
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Population Correlation Coefficients
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Standard Deviation Matrix
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Correlation Matrix from Covariance Matrix
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Partitioning Covariance Matrix
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Partitioning Covariance Matrix
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Linear Combinations of Random Variables
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Example of Linear Combinations of Random
Variables
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Linear Combinations of Random Variables
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Sample Mean Vector and Covariance Matrix
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Partitioning Sample Mean Vector
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Partitioning Sample Covariance Matrix
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Cauchy-Schwarz Inequality
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Extended Cauchy-Schwarz Inequality
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Maximization Lemma
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Maximization of Quadratic Forms for Points on the
Unit Sphere
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Maximization of Quadratic Forms for Points on the
Unit Sphere
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Maximization of Quadratic Forms for Points on the
Unit Sphere
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Maximization of Quadratic Forms for Points on the
Unit Sphere