04. Matrix Properties

Determinant

A determinant is a scalar number associated to a square matrix.

The determinant is the scale factor of the transformation A.

1x1

The matrix:

A:=\begin{bmatrix} a \end{bmatrix}

... has the determinant:

\det A=a

2x2

The matrix:

A:=\begin{bmatrix} a & b\\c & d \end{bmatrix}

... has the determinant:

\det A=a⋅d-b⋅c

3x3

Use the Rule of Sarrus to find the determinant.

More

Develop the determinant by finding all the permutations.

The rule of Sarrus doesn't work for matrices bigger than 3x3.

Simple cases

If the matrix is a triangular matrix, the determinant is just the product of the diagonal entries.

If the rank of the matrix is less than the number of columns, the determinant equals 0.

determinant=+1|-1: unimodular.

determinant=0: singular matrix, not invertible.

Invariants

\det{A}=\det{A^T}
\det{I}=1

Algebra with determinants

\det(A⋅B)=(\det A)⋅(\det B)
\det(λ⋅A)=λ^n⋅\det A

Definiteness

Quadratic form

A quadratic form (for a body of real numbers) is:

q(x⃗):=x⃗^T⋅A⋅x⃗

A matrix A is positiv definite iff q(x⃗)>0∀x⃗≠0⃗.

A matrix A is negativ definite iff q(x⃗)<0∀x⃗≠0⃗.

Hauptminoren

A matrix A is positiv definite iff all Hauptminors of A are >0.

A matrix A is negativ definite iff all Hauptminors of -A are >0.

A Hauptminor of A is the determinant of a matrix which has only the first n⨯n cells of the original matrix A.

Eigenvalues

An eigenvalue of A is the solution of the following equation for λ:

A⋅v⃗=λ⋅v⃗
(A-λ⋅I)⋅v⃗=0⃗
\det (A-λ⋅I)=0

A matrix A is positiv definite iff all eigenvalues are > 0.

A matrix A is negativ definite iff all eigenvalues are < 0.

A matrix A is indefinite iff there exist eigenvalues which are < 0 and eigenvalues which are > 0.

Hauptachsentransformation

See Hauptachsentransformation.

Norm

The norm of the matrix A is defined as:

||A||:=\max_{||v⃗||=1} ||A⋅v⃗||

That is, it is the maximum scale factor of the matrix A.

The following inequality holds:

||A⋅x⃗||≤||A||⋅||x⃗||∀x⃗

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Last modification on: Sat, 04 May 2024 .