Difference between revisions of "The transpose of a matrix"
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{{Navigation|before=[[Multiplication with a scalar]]|overview=[[Matrices]]|next=[[Addition of matrices]]}} | {{Navigation|before=[[Multiplication with a scalar]]|overview=[[Matrices]]|next=[[Addition of matrices]]}} | ||
| + | {{Exercise|Selftest: Transpose}} | ||
The transpose of an m-by-n matrix <math>\mathbf{A}</math> is the n-by-m matrix <math>\mathbf{A}^T</math> where all the colums are written as rows and all the rows as colums: <br/><br/> | The transpose of an m-by-n matrix <math>\mathbf{A}</math> is the n-by-m matrix <math>\mathbf{A}^T</math> where all the colums are written as rows and all the rows as colums: <br/><br/> | ||
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(\mathbf{A} \cdot \mathbf{B})^T &= \mathbf{B}^T \cdot \mathbf{A}^T\\ | (\mathbf{A} \cdot \mathbf{B})^T &= \mathbf{B}^T \cdot \mathbf{A}^T\\ | ||
\left(\mathbf{A}^{-1}\right)^T &= \left(\mathbf{A}^T\right)^{-1} | \left(\mathbf{A}^{-1}\right)^T &= \left(\mathbf{A}^T\right)^{-1} | ||
| − | \end{align}</math | + | \end{align}</math> |
{{Example | {{Example | ||
Latest revision as of 18:03, 13 November 2015
| ← Back: Multiplication with a scalar | Overview: Matrices | Next: Addition of matrices → |
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There are exercises as selftest for this article. |
The transpose of an m-by-n matrix 
is the n-by-m matrix 
where all the colums are written as rows and all the rows as colums:
If an n-by-n square matrix is concerned the transpose is obtained by mirroring the matrix at its first diagonal from top left to bottom right.
In general the following arithmetic rules hold:
 Example: The transpose of a matrix
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