Difference between revisions of "Transposed matrix"
From Encyclopedia of Mathematics
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| − | The [[Matrix|matrix]] obtained from a given (rectangular or square) matrix | + | {{TEX|done}} |
| + | The [[Matrix|matrix]] obtained from a given (rectangular or square) matrix $A=\|a_{ik}\|$ ($i=1,\dots,m$; $k=1,\dots,n$) by interchanging the rows and the columns, that is, the matrix $\|a_{ik}'\|$, where $a_{ik}'=a_{ki}$ ($i=1,\dots,n$; $k=1,\dots,m$). The number of rows of the transposed matrix is equal to the number of columns of $A$, while the number of columns is equal to the number of rows of $A$. The transpose of a matrix $A$ is usually denoted by $A^T$ or $A'$. | ||
====Comments==== | ====Comments==== | ||
| − | Some elementary properties of the transposition of matrices are | + | Some elementary properties of the transposition of matrices are $(A+B)^T=A^T+B^T$, $(\alpha A)^T=\alpha A^T$, $(AB)^T=B^TA^T$, $(A^{-1})^T=(A^T)^{-1}$. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1959) pp. 19 (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1959) pp. 19 (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 16:07, 4 October 2014
The matrix obtained from a given (rectangular or square) matrix $A=\|a_{ik}\|$ ($i=1,\dots,m$; $k=1,\dots,n$) by interchanging the rows and the columns, that is, the matrix $\|a_{ik}'\|$, where $a_{ik}'=a_{ki}$ ($i=1,\dots,n$; $k=1,\dots,m$). The number of rows of the transposed matrix is equal to the number of columns of $A$, while the number of columns is equal to the number of rows of $A$. The transpose of a matrix $A$ is usually denoted by $A^T$ or $A'$.
Comments
Some elementary properties of the transposition of matrices are $(A+B)^T=A^T+B^T$, $(\alpha A)^T=\alpha A^T$, $(AB)^T=B^TA^T$, $(A^{-1})^T=(A^T)^{-1}$.
References
| [a1] | F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1959) pp. 19 (Translated from Russian) |
How to Cite This Entry:
Transposed matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transposed_matrix&oldid=15848
Transposed matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transposed_matrix&oldid=15848
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article