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3. Numerical Linear Algebra 97

T
T
⇒X a = X (Xs)

T
T
-1
⇒s=(X X) X a

T
-1
T
Therefore the projected vector p=[X][s]=X(X X) X a

-1
T
T
Thus projected matrix P M is defined as X(X X) X .Also the projected
vector p is represented as [P M] a
In general projection of the vector ‘a’ on the vector space spanned by the
T
T
-1
column vectors of the matrix ‘A’ is given as A(A A) A a

4.2.1 Example
Consider the vector space spanned by the column vectors of the matrix

A= 1 3 7 6

4 6 2 4

1 7 6 4

T
The projection of the vector a= [1 2 3]

on the vector space spanned by the column vectors of the matrix A is
T
T
-1
given using the formula A(A A) A a is given below

Projection matrix

T
-1
T
P M = A(A A) A =

-0.2813 -0.5625 -0.6563
-0.1875 1.1250 0.0625
0.0313 0.1875 1.5313