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| = Projections = | = Projection = |
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| When two vectors do not exist in the same column space, the best approximation of one in the other's columns space is called a '''projection'''. | A '''projection''' is an approximation within a column space. A '''projection matrix''' describes the linear transformation of the approximation. See also [[Calculus/Projection|vector projection]]. |
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| Given two vectors ''a'' and ''b'', we can '''project''' ''b'' onto ''a'' to get the best possible estimate of the former as a multiple of the latter. This projection ''p'' has an error term ''e''. | Given vectors ''a'' and ''b'', ''b'' can be [[Calculus/Projection|projected]] into the column space of ''a'' (i.e., ''C(a)''). A '''projection matrix''' describes the linear transformation from vector ''a'' to projection vector ''p''. Such a projection matrix '''''P''''' satisfies ''p = '''P'''a''. |
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| Take the multiple as ''x'', so that ''p = ax''. The error term can be characterized as ''b-p'' or ''b-ax''. | The column space of the projection matrix (i.e., ''C('''P''')'') is equivalent to ''C(b)''. It follows that '''''P''''' is also of [[LinearAlgebra/Rank|rank]] 1. |
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| ''a'' is [[LinearAlgebra/Orthogonality|orthogonal]] to ''e''. Therefore, ''a^T^(b-ax) = 0''. This simplifies to ''x = (a^T^b)/(a^T^a)''. Altogether, the projection is characterized as ''p = a(a^T^b)/(a^T^a)''. | |
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| A matrix '''''P''''' can be defined such that ''p = '''P'''b''. The projection matrix is ''(aa^T^)/(a^T^a)''. The column space of '''''P''''' (a.k.a. ''C('''P''')'') is the line through ''a'', and its rank is 1. | |
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| Incidentally, '''''P''''' is symmetric (i.e. '''''P'''^T^ = '''P''''') and re-projecting does not change the result (i.e. '''''P'''^2^ = '''P'''''). | === Properties === Projection matrices are [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric]] (i.e., '''''P'''^T^ = '''P''''') and [[LinearAlgebra/Idempotency|idempotent]] (i.e., '''''P'''^2^ = '''P'''''). |
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| For problems like '''''A'''x = b'' where there is no solution for ''x'', as in b does not exist in the column space of '''''A''''', we can instead solve '''''A'''x = p'' where ''p'' estimates ''b'' with an error term ''e''. | Given vector ''b'' and matrix '''''A''''', ''b'' can be projected into the column space of '''''A''''' (i.e., ''C('''A''')''). |
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| ''p'' is a linear combination of '''''A''''': if there are two columns ''a,,1,,'' and ''a,,2,,'', then ''p = x,,1,,a,,1,, + x,,2,,a,,2,,'' and ''b = x,,1,,a,,1,, + x,,2,,a,,2,, + e''. | Given a linear system as '''''A'''x = b'', if ''b'' is in ''C('''A''')'', there are solutions for ''x''. If ''b'' is ''not'' in ''C('''A''')'' however, there is no possible solution. The best ''approximation'' is expressed as '''''A'''x̂ = p'' where projection ''p'' estimates ''b'' with an error term ''e''. [[Statistics/OrdinaryLeastSquares|This should sound familiar.]] |
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| ''e'' is orthogonal to the column space of '''''A'''^T^'' (a.k.a. ''C('''A'''^T^)''), so '''''A'''^T^(b-'''A'''x) = 0''. Concretely in the same example, ''a,,1,,^T^(b-'''A'''x) = 0'' and ''a,,2,,^T^(b-'''A'''x) = 0''. More generally, this re-emphasizes that ''e'' is orthogonal in the null space of '''''A'''^T^'' (a.k.a. ''N('''A'''^T^)''). | The error term can be characterized by ''e = b - p'' or ''e = b - '''A'''x̂''. ''e'' is orthogonal to ''R('''A''')'', the row space of '''''A'''''; equivalently it is orthogonal to ''C('''A'''^T^)''. Orthogonality in this context means that ''e'' is in the [[LinearAlgebra/NullSpace|null space]], so '''''A'''^T^(b - '''A'''x̂) = 0''. |
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| The solution for this all is ''x = ('''A'''^T^'''A''')^-1^'''A'''^T^b''. That also means that ''p = '''A'''('''A'''^T^'''A''')^-1^'''A'''^T^b''. | The system of '''normal equations''' is '''''A'''^T^'''A'''x̂ = '''A'''^T^b''. This simplifies to ''x̂ = ('''A'''^T^'''A''')^-1^'''A'''^T^b''. Altogether, the projection is characterized by ''p = '''A'''('''A'''^T^'''A''')^-1^'''A'''^T^b''. |
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| A matrix '''''P''''' can be defined such that ''p = '''P'''b''. The projection matrix is '''A'''('''A'''^T^'''A''')^-1^'''A'''^T^. | The projection matrix '''''P''''' satisfies ''p = '''P'''b''. It is calculated as '''''P''' = '''A'''('''A'''^T^'''A''')^-1^'''A'''^T^''. |
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| Note that if '''''A''''' were a square matrix, most of the above equations would [[LinearAlgebra/MatrixInversion|cancel out]]. But we cannot make that assumption. This fundamentally means though that if ''b'' were in the column space of '''''A''''', then '''''P''''' would be the identity matrix. | ''b'' can also be projected onto ''e'', which geometrically means projecting into the null space of '''''A'''^T^''. Algebraically, if one projection matrix has been computed as '''''P''''', then the projection matrix for going the other way is ''('''I''' - '''P''')b''. |
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| [[Econometrics/OrdinaryLeastSquares|This should look familiar.]] | === Properties === As above, the projection matrix '''''P''''' is symmetric and idempotent. If '''''A''''' is square, the above equations simplify rapidly. If ''b'' is in ''C('''A''')'', then '''''P''' = '''I'''''. Conversely, if ''b'' is orthogonal to ''C('''A''')'', then '''''P'''b = 0'' and ''b = e''. |
Projection
A projection is an approximation within a column space. A projection matrix describes the linear transformation of the approximation.
See also vector projection.
Contents
Vectors
Given vectors a and b, b can be projected into the column space of a (i.e., C(a)). A projection matrix describes the linear transformation from vector a to projection vector p. Such a projection matrix P satisfies p = Pa.
The column space of the projection matrix (i.e., C(P)) is equivalent to C(b). It follows that P is also of rank 1.
Properties
Projection matrices are symmetric (i.e., PT = P) and idempotent (i.e., P2 = P).
Matrices
Given vector b and matrix A, b can be projected into the column space of A (i.e., C(A)).
Given a linear system as Ax = b, if b is in C(A), there are solutions for x. If b is not in C(A) however, there is no possible solution. The best approximation is expressed as Ax̂ = p where projection p estimates b with an error term e. This should sound familiar.
The error term can be characterized by e = b - p or e = b - Ax̂. e is orthogonal to R(A), the row space of A; equivalently it is orthogonal to C(AT). Orthogonality in this context means that e is in the null space, so AT(b - Ax̂) = 0.
The system of normal equations is ATAx̂ = ATb. This simplifies to x̂ = (ATA)-1ATb. Altogether, the projection is characterized by p = A(ATA)-1ATb.
The projection matrix P satisfies p = Pb. It is calculated as P = A(ATA)-1AT.
b can also be projected onto e, which geometrically means projecting into the null space of AT. Algebraically, if one projection matrix has been computed as P, then the projection matrix for going the other way is (I - P)b.
Properties
As above, the projection matrix P is symmetric and idempotent.
If A is square, the above equations simplify rapidly.
If b is in C(A), then P = I. Conversely, if b is orthogonal to C(A), then Pb = 0 and b = e.