25. Symmetric Matrices and Positive Definiteness

Overview

Real symmetric matrices have real eigenvalues and admit an orthonormal eigenbasis, yielding the spectral factorization A = QΛQᵀ. The lecture connects this structure to rank-one projection matrices, explains how pivot signs reveal eigenvalue signs, and introduces positive definite matrices through positive eigenvalues, positive pivots, and positive leading principal minors.

Key takeaways

Every real symmetric matrix has real eigenvalues and an orthonormal eigenbasis, so it can be diagonalized as A = QΛQᵀ with Q⁻¹ = Qᵀ.
For a complex eigenvector x, the quantity x̄ᵀx = Σ|xᵢ|² is positive when x is nonzero; this makes it possible to prove that symmetric-matrix eigenvalues are real.
The spectral decomposition A = Σ λᵢqᵢqᵢᵀ represents a symmetric matrix as a weighted sum of mutually orthogonal rank-one projection matrices.
For symmetric matrices, pivot signs reveal the counts of positive and negative eigenvalues, avoiding the need to compute every eigenvalue directly.
A symmetric matrix is positive definite exactly when its eigenvalues are positive; Sylvester’s criterion checks this by requiring every leading principal minor to be positive.
For [[5, 2], [2, 3]], the pivots 5 and 11/5, leading principal minors 5 and 11, and eigenvalues 4 ± √5 all confirm positive definiteness.

Chapters

0:00 Real Eigenvalues and Orthogonal Eigenvectors of Symmetric Matrices

A real symmetric matrix satisfies A = Aᵀ and has only real eigenvalues.
Eigenvectors for distinct eigenvalues are perpendicular; for repeated eigenvalues, an orthonormal basis can be chosen within the eigenspace.
The identity matrix illustrates the repeated-eigenvalue case: every vector is an eigenvector, so an orthogonal basis must be selected.

8:00 The Spectral Theorem: A = QΛQᵀ

Scaling the eigenvectors to unit length forms an orthogonal matrix Q, whose columns are orthonormal eigenvectors.
Since Q⁻¹ = Qᵀ, a symmetric matrix factors as A = QΛQᵀ, with real eigenvalues on the diagonal of Λ.
The factorization is called the spectral theorem and is also known in mechanics as the principal axis theorem.

9:00 Why a Real Symmetric Matrix Has Real Eigenvalues

Starting from Ax = λx, take complex conjugates and use that A is real to obtain A x̄ = λ̄ x̄.
Transpose the conjugated equation and use Aᵀ = A to compare it with the original eigenvalue equation.
The comparison leads to λ x̄ᵀx = λ̄ x̄ᵀx, so showing x̄ᵀx is nonzero establishes λ = λ̄.

17:00 Complex Inner Products and the Hermitian Generalization

For a complex vector x, x̄ᵀx = Σ|xᵢ|² is real and strictly positive when x is nonzero, allowing the proof to cancel this quantity.
A real matrix has nonreal eigenvalues in conjugate pairs, while symmetry rules out those pairs by forcing each eigenvalue to be real.
For complex matrices, the corresponding condition is Āᵀ = A, the Hermitian property.

24:30 Decomposing Symmetric Matrices into Projection Matrices

Writing the orthonormal eigenvectors as qᵢ gives A = Σ λᵢ qᵢqᵢᵀ.
Each qᵢqᵢᵀ is a symmetric projection matrix because qᵢᵀqᵢ = 1 and (qᵢqᵢᵀ)² = qᵢqᵢᵀ.
The spectral theorem therefore expresses every real symmetric matrix as a weighted sum of mutually orthogonal projections.

29:00 Using Pivot Signs to Count Positive and Negative Eigenvalues

For symmetric matrices, symmetric elimination connects the signs of the pivots with the counts of positive and negative eigenvalues.
Counting pivots can be more practical than computing every eigenvalue of a 50-by-50 or 100-by-100 matrix.
Shifting A by 7I shifts each eigenvalue by 7, so pivot-sign counts for A − 7I reveal how many original eigenvalues lie above or below 7.
The pivot product and eigenvalue product both equal the determinant, but matching sign counts provides information beyond that product.

35:00 Positive Definiteness and the Matrix [[5, 2], [2, 3]]

A positive definite matrix is symmetric and has only positive eigenvalues; equivalently, its symmetric-elimination pivots are positive.
For [[5, 2], [2, 3]], the pivots are 5 and 11/5, both positive, while the determinant is 11.
The characteristic polynomial is λ² − 8λ + 11, giving eigenvalues 4 ± √5, which are both positive.
Positive definiteness is useful in differential equations because eigenvalue signs help determine stability.

39:50 Leading Principal Minors and the Positive-Definite Test

A positive determinant alone does not guarantee positive definiteness: a matrix can have positive determinant while its first pivot is negative.
Sylvester’s criterion tests positive definiteness by checking that every leading principal minor is positive, from the upper-left 1-by-1 minor through the full determinant.
For [[5, 2], [2, 3]], the leading principal minors are 5 and 11, so both pass the test.
The lecture brings together pivots, determinants, and eigenvalues as complementary ways to characterize symmetric positive definite matrices.

Transcript excerpt

0:00 -- one and -- the lecture on symmetric matrixes. So that's the most important class of matrixes, symmetric matrixes. A equals A transpose. So the first points, the main points of the lecture I'll tell you right away. What's special about the eigenvalues? What's special about the eigenvectors? This is -- the way we now look at a matrix. We want to know about its eigenvalues and eigenvectors and if we have a special type of matrix,

0:48 that should tell us something about eigenvalues and eigenvectors. Like Markov matrixes, they have an eigenvalue equal one. Now symmetric matrixes, can I just tell you right off what the main facts -- the two main facts are? The eigenvalues of a symmetric matrix, real -- this is a real symmetric matrix, we -- talking mostly about real matrixes. The eigenvalues are also real. So our examples of rotation matrixes, where -- where we got E- eigenvalues that were complex, that won't happen now. For symmetric matrixes, the eigenvalues are real and the eigenvectors are also very special.

1:38 The eigenvectors are perpendicular, orthogonal, so which do you prefer? I'll say perpendicular. Perp- well, they're both long words. Okay, right. So -- I have a -- you should say "why?" and I'll at least answer why for case one, maybe case two, the checking the Eigen -- that the eigenvectors are perpendicular, I'll leave to, the -- to the book. But let's just realize what -- well, first I have to say, it -- it could happen, like for the identity matrix --

2:29 there's a symmetric matrix. Its eigenvalues are certainly all real, they're all one for the identity matrix. What about the eigenvectors? Well, for the identity, every vector is an eigenvector. So how can I say they're perpendicular? What I really mean is the -- they --

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