Positive definite matrices: every eigenvalue positive
Slide 5 showed that a symmetric matrix stretches along perpendicular axes, each by its own eigenvalue. Now ask for the nicest case: every one of those stretches is positive, so no axis is flipped. Such a matrix is called positive definite. Think of a marble. In a bowl it rolls to the one lowest point, wherever you drop it. On a horse saddle it rolls off, because one direction goes up and another goes down. For a symmetric A, the height over the point x is the number xᵀAx. Positive definite matrices give bowls: xᵀAx > 0 for every x except 0. One negative eigenvalue gives a saddle.
Lesson 7 · slide 6Positive definite- A symmetric A is positive definite when all its eigenvalues are positive.
- Equivalent test: the energy xᵀAx (defined below) > 0 for every nonzero x.
- For 2 by 2 abbc: a > 0 and ac − b² > 0. (Also: all pivots positive.)
- The graph of xᵀAx is a bowl. One negative eigenvalue turns it into a saddle.
The number xᵀAx
xᵀAx is the dot product of x with Ax: one number for every x, often called the energy. The name comes from springs: for the row of springs in the bigger example below, ½xᵀAx is the energy stored when the masses are moved by x. For A = abbc and a point x = xy, writing x and y for its two entries:
It is a sum of squares, so it is positive unless x = 0. Read it as a surface: for every point of the floor, raise a dot to height xᵀAx; the dots form a surface. Here that surface is a bowl with its lowest point at the origin. Since xᵀAx = x · Ax, being positive also means Ax never points more than 90° away from x (lesson 1: a positive dot product means an angle under 90°).
The eigenvalue test
Write x in the perpendicular unit eigenvectors from slide 5: x = k₁q₁ + k₂q₂. Then Ax = λ₁k₁q₁ + λ₂k₂q₂. Multiply out the dot product of the two, using q₁ · q₁ = q₂ · q₂ = 1 and q₁ · q₂ = 0:
Both λ positive: this is positive for every nonzero x. A bowl. One λ negative: move along its eigenvector and xᵀAx goes negative. A saddle. Both λ negative: xᵀAx is negative for every nonzero x, an upside-down bowl (a hilltop).
Quick tests for 2 by 2
abbc is positive definite exactly when a > 0 and ac − b² > 0. The determinant ac − b² equals λ₁λ₂ (slide 2), so a negative determinant means one λ is negative: a saddle. A positive determinant means both λ have the same sign. Why also a > 0? Because a is xᵀAx at x = 10, so a > 0 rules out both λ being negative (the upside-down bowl). For 2bb2, the determinant 4 − b² is negative (a saddle) when b² > 4.
The pivot test
Elimination gives the same answer: A is positive definite exactly when all its pivots (lesson 2) are positive. For a 2 by 2 matrix the pivots are a and (ac − b²)/a, so this is the same test as above:
Pivots are the cheap test for big matrices: one pass of elimination, no eigenvalues needed.
Keep reading with the full course
Finish lesson 7 and you will be able to: tell whether a system settles down or blows up.
Still locked here:
- The rest of this explanation
- 4 more examples with step-by-step solutions
- Five extra members-only problems with worked solutions
$20 once, lifetime access. Lessons 1 to 3 stay free.
Examples
Example 1 · A diagonal bowl
A = 3001: xᵀAx = 3x² + y², positive for x ≠ 0. λ = 3 and 1.
Questions and comments
Stuck on something, or found a mistake? Ask here and I'll answer. You sign in with GitHub.