#Type/Definition #Topic/Linear_Algebra #Type/Example

Types: N/A
Examples: 9.12.1 Example of Gauss Transform in Action
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: 8.7 Lower-Triangular Matrix
Questions: N/A

Gauss Transform

Let natural numbers n,k∈N with k<n. Let 3.1 Column Vector τ→tau∈Rn whose first k components are zero. Suppose τ→ is in the form

τ→T=[0…0⏟kτk+1…τn]

Then, a Gauss transformation is a matrix

Lk=In−τ→⋅e→kT

We call the vector τ→ a Gauss vector.

Gauss Transform

Let's look at a Gauss transform in the case of n=5 and k=2. Let

τ→=[002−14]

Let's compute.

L2=Lk=In−τ→⋅e→kT=I5−τ→⋅e→2T=[1000001000001000001000001]−[002−14]⋅[01000]=[1000001000001000001000001]−[0000000000020000−100004000]=[10000010000−2100010100−4001]

The resulting matrix is a 8.8 Unit Lower-Triangular Matrix. We can also rewrite this as the product of three shear matrices.

⟹L2=S52(−4)⋅S42(1)⋅S32(−2)=[100000100000100000100−4000]⋅[1000001000001000101000001]⋅[1000001000021000001000001]