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Unit 8: Symmetric Matrices and Positive Definiteness

Why symmetric matrices always diagonalize using perpendicular directions alone, and what makes a quadratic bowl curve upward.

6 topics · 22 questions

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Symmetric Matrices and Real Eigenvalues

  1. How it worksHow does A equal A transpose guarantee that every eigenvalue of A comes out as a real number?
  2. Worked exampleFind the eigenvalues of the symmetric matrix [[2, 1], [1, 2]].
  3. Common mistakeWhy doesn't a matrix being symmetric mean its entries are all equal to each other?
  4. Going furtherWhat happens to the eigenvectors of a symmetric matrix that has two different eigenvalues?
Your turn1 of 2
Find the LARGER eigenvalue of the symmetric matrix [[2, 4], [4, 5]].

The Spectral Theorem

  1. How it worksHow does the spectral theorem write a symmetric matrix as Q Lambda Q transpose using an orthogonal Q?
  2. Worked exampleFind the orthogonal diagonalization of the matrix [[3, 1], [1, 3]].
  3. Common mistakeWhy doesn't every square matrix admit this same clean Q Lambda Q transpose decomposition?
  4. Ties togetherHow does the spectral theorem connect the diagonalization seen earlier to orthogonal matrices specifically?
Your turn1 of 2
Find the LARGER eigenvalue of the symmetric matrix [[7, 4], [4, 8]], as part of its orthogonal diagonalization.

Orthogonal Diagonalization

  1. How it worksHow does normalizing each eigenvector of a symmetric matrix build the orthogonal matrix Q?
  2. Worked exampleFind an orthonormal eigenbasis for the matrix [[5, 4], [4, 5]].
  3. Common mistakeWhy doesn't orthogonal diagonalization work for a matrix that isn't symmetric?
Your turn1 of 2
Find the SMALLER eigenvalue of the symmetric matrix [[7, 4], [4, 8]].

Positive Definite Matrices

  1. How it worksHow does every eigenvalue of a matrix turning out positive guarantee the matrix is positive definite?
  2. Worked exampleShow that the matrix [[2, 0], [0, 3]] is positive definite by checking its eigenvalues.
  3. Common mistakeWhy doesn't having all positive entries make a matrix positive definite?
  4. Going furtherWhat happens to positive definiteness once a positive definite matrix gets multiplied by a negative number?
Your turn1 of 2
Find the determinant of the positive definite matrix [[7, 0], [0, 6]] — the product of its eigenvalues.

Tests for Positive Definiteness

  1. How it worksHow does checking that every leading principal minor is positive test for positive definiteness?
  2. Worked exampleFind whether the matrix [[2, 1], [1, 2]] is positive definite using its leading principal minors.
  3. Going furtherWhat happens to positive definiteness the moment one diagonal entry of a matrix turns negative?
Your turn1 of 2
Find the value of the leading 2 by 2 principal minor (determinant) for the matrix [[7, 2], [2, 8]], used to test positive definiteness.

Quadratic Forms and Their Graphs

  1. How it worksHow does writing a quadratic expression as x transpose A x connect it back to a symmetric matrix?
  2. Worked exampleFind the matrix of the quadratic form 3x2 + 4xy + 3y2.
  3. Common mistakeWhy doesn't a quadratic form with all positive coefficients guarantee its graph is a bowl opening upward?
  4. Ties togetherHow does the sign of a quadratic form's eigenvalues decide whether its graph is a bowl, a saddle, or a dome?
Your turn1 of 2
Find the (1,2) entry of the symmetric matrix associated with the quadratic form 3x2 + 2xy + 7y2.