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== Transpose of linear maps == |
== Transpose of linear maps == |
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If ''f'': V→W is a [[linear operator|linear map]] between [[vector space]]s V and W with [[ |
If ''f'': V→W is a [[linear operator|linear map]] between [[vector space]]s V and W with [[nondegenerate]] [[bilinear form]]s, we define the ''transpose'' of ''f'' to be the linear map <sup>t</sup>''f'' : W→V determined by |
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:<math>B_V(v,{}^tf(w))=B_W(f(v),w)</math> |
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:<div style="vertical-align: 10%;display:inline;"><math> |
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for all ''v'' in ''V'' and ''w'' in ''W''. Here, ''B''<sub>''V''</sub> and ''B''<sub>''W''</sub> are the bilinear forms on ''V'' and ''W'' respectively. The matrix of the transpose of a map is the transposed matrix only if the [[basis (linear algebra)|bases]] are orthonormal with respect to their bilinear forms. |
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{}^t f (\phi ) = \phi \circ f \,</math></div> for every <math>\ \phi</math> in W*. |
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Over a complex vector space, one often works with [[sesquilinear form]]s instead of bilinear (conjugate-linear in one argument). The transpose of a map between such spaces is defined similarly, and the matrix of the transpose map is given by the conjugate transpose matrix if the bases are orthonormal. |
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If the matrix ''A'' describes a linear map with respect to two [[basis (linear algebra)|bases]], then the matrix ''A''<sup>T</sup> describes the transpose of that linear map with respect to the dual bases. See [[dual space]] for more details on this. |
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[[Category:Abstract algebra]] |
[[Category:Abstract algebra]] |
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Revision as of 02:17, 7 March 2006
- See transposition for meanings of this term in telecommunication and music.
In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. Informally, the transpose of a square matrix is obtained by reflecting at the main diagonal (that runs from the top left to bottom right of the matrix). The transpose of the matrix A is written as Atr, tA, A′, or AT.
Formally, the transpose of the m-by-n matrix A is the n-by-m matrix AT defined by AT[i, j] = A[j, i] for 1 ≤ i ≤ n and 1 ≤ j ≤ m.
For example,
Properties
For any two m-by-n matrices A and B and every scalar c, we have (A + B)T = AT + BT and (cA)T = c(AT). This shows that the transpose is a linear map from the space of all m-by-n matrices to the space of all n-by-m matrices.
The transpose operation is self-inverse, i.e taking the transpose of the transpose amounts to doing nothing: (AT)T = A.
If A is an m-by-n and B an n-by-k matrix, then we have (AB)T = (BT)(AT). Note that the order of the factors switches. From this one can deduce that a square matrix A is invertible if and only if AT is invertible, and in this case we have (A-1)T = (AT)-1.
The dot product of two vectors expressed as columns of their coordinates can be computed as
where the product on the right is the ordinary matrix multiplication.
If A is an arbitrary m-by-n matrix with real entries, then ATA is a positive semidefinite matrix.
If A is an n-by-n matrix over some field, then A is similar to AT.
Further nomenclature
A square matrix whose transpose is equal to itself is called a symmetric matrix, i.e. A is symmetric iff:
A square matrix whose transpose is also its inverse is called an orthogonal matrix, i.e. G is orthogonal iff
- the identity matrix
A square matrix whose transpose is equal to its negative is called skew-symmetric, i.e. A is skew-symmetric iff:
The conjugate transpose of the complex matrix A, written as A*, is obtained by taking the transpose of A and then taking the complex conjugate of each entry.
Transpose of linear maps
If f: V→W is a linear map between vector spaces V and W with nondegenerate bilinear forms, we define the transpose of f to be the linear map tf : W→V determined by
for all v in V and w in W. Here, BV and BW are the bilinear forms on V and W respectively. The matrix of the transpose of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms.
Over a complex vector space, one often works with sesquilinear forms instead of bilinear (conjugate-linear in one argument). The transpose of a map between such spaces is defined similarly, and the matrix of the transpose map is given by the conjugate transpose matrix if the bases are orthonormal.