Symmetric Matrices and Quadratic Forms

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Related lectures (28)

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Characteristic Polynomials and Similar Matrices

Explores characteristic polynomials, similarity of matrices, and eigenvalues in linear transformations.

Spectral Theorem Recap

Revisits the spectral theorem for symmetric matrices, emphasizing orthogonally diagonalizable properties and its equivalence with symmetric bilinear forms.

Quadratic Forms and Symmetric Bilinear Forms

Explores quadratic forms, symmetric bilinear forms, and their properties.

Sylvester's Inertia Theorem

Explores Sylvester's Inertia Theorem, relating eigenvalues to diagonal entries in symmetric matrices.

Non-Negative Definite Matrices and Covariance Matrices

Covers non-negative definite matrices, covariance matrices, and Principal Component Analysis for optimal dimension reduction.

Quadratic Forms and Symmetric Matrices

Explores examples of algebraic quotients using invariance maps and discriminant.

Orthogonal Projection Theorems

Covers the theorems related to orthogonal projection and orthonormal bases.

Linear Algebra: Quadratic Forms and Matrix Diagonalization

Discusses quadratic forms, matrix diagonalization, and their applications in optimization problems.

Diagonalization of Symmetric Matrices

Explores the diagonalization of symmetric matrices through orthogonal decomposition and the spectral theorem.

Matrix Diagonalization: Spectral Theorem

Covers the process of diagonalizing matrices, focusing on symmetric matrices and the spectral theorem.