Symmetric, Positive Definite Matrix

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Table of contents
  1. Relation to Inner Product
    1. Symmetric, Positive Semi-Definite Matrix
  2. Nullspace of Symmetric, Positive Definite Matrix
  3. Diagonal Elements of Symmetric, Positive Definite Matrix

Relation to Inner Product

Any general inner product can be represented as a matrix multiplication.

The matrix that uniquely defines the inner product is called the symmetric, positive definite matrix.

Let V be an n-dimensional inner product space with inner product ,:V×VR.

Let B=(b1,,bn) be an ordered basis of V.

Any vector x,yV can be represented as a linear combination of the basis vectors:

x=i=1nλibiy=j=1nψjbj

for some λi,ψjR.

Also, let x^=[λ1λn]T and y^=[ψ1ψn]T, which are the coordinate vectors of x and y with respect to the ordered basis B.

Often also denoted [x]B and [y]B. See coordinate vectors with respect to orderded bases.

The inner product x,y have three properties:

First using bilinearity:

x,y=i=1nλibi,j=1nψjbj=i=1nj=1nλibi,bjψj=x^TAy^

x^,y^ are just coordinates, so A is the matrix that uniquely defines the inner product operation.

Because an inner product is symmetric, A is also a symmetric matrix.

Now using positive definiteness:

(Positive Definite)xV{0},xTAx>0

Such A is called the symmetric, positive definite matrix.

Hence, x,y is a valid inner product if and only if there exists a symmetric, positive definite matrix ARn×n such that

x,y=x^TAy^

Again, remember that x^ and y^ are coordinate vectors of each respect to the ordered basis B.

Symmetric, Positive Semi-Definite Matrix

In equation (Positive Definite), if the inequality is loosened to , then A is called the symmetric, positive semi-definite matrix.


Nullspace of Symmetric, Positive Definite Matrix

By definition, we know that xTAx>0 for all x0. So trivially, Ax=0 only when x=0.

Therefore, the kernel of A only contains the zero vector.

In other words, the columns of A are linearly independent. Also, A is invertible.


Diagonal Elements of Symmetric, Positive Definite Matrix

Diagonal elements of A are positive, because aii=eiTAei>0.