Symmetric, Positive Definite Matrix
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Table of contents
- Relation to Inner Product
- Symmetric, Positive Semi-Definite Matrix
- Nullspace of Symmetric, Positive Definite Matrix
- 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 be an -dimensional inner product space with inner product .
Let be an ordered basis of .
Any vector can be represented as a linear combination of the basis vectors:
for some .
Also, let and , which are the coordinate vectors of and with respect to the ordered basis .
Often also denoted and . See coordinate vectors with respect to orderded bases.
The inner product have three properties:
First using bilinearity:
are just coordinates, so is the matrix that uniquely defines the inner product operation.
Because an inner product is symmetric, is also a symmetric matrix.
Now using positive definiteness:
Such is called the symmetric, positive definite matrix.
Hence, is a valid inner product if and only if there exists a symmetric, positive definite matrix such that
Again, remember that and are coordinate vectors of each respect to the ordered basis .
Symmetric, Positive Semi-Definite Matrix
In equation , if the inequality is loosened to , then is called the symmetric, positive semi-definite matrix.
Nullspace of Symmetric, Positive Definite Matrix
By definition, we know that for all . So trivially, only when .
Therefore, the kernel of only contains the zero vector.
In other words, the columns of are linearly independent. Also, is invertible.
Diagonal Elements of Symmetric, Positive Definite Matrix
Diagonal elements of are positive, because .