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The transpose of a matrix: Courses and properties

This article aims to present the transpose of a matrix through its definition, properties and examples. It is good to have the basic knowledge of what a matrix is.

Definition

Let A be a matrix (not necessarily square) of size nxp defined by its coefficients (aij). The transpose A, noted tA is the matrix whose symmetry is made with respect to the direct diagonal. It is therefore a matrix of size px n. Its coefficient i,j is defined by

\forall i \in \{1, \ldots, p \}, \forall j \in \{1, \ldots, n\},(^tA) _{ij}= a_{ji}

It is therefore an application of

M_{n,p}(\mathbb K) \mapsto M_{p,n}(\mathbb K)

Warning: it can have several notations. It can for example be noted on the right and with a capital T:

A^T

Examples

Example 1: With a square matrix

Take the following matrix:

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}

The transpose of A is then

^tA = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8\\ 3 & 6 & 9 \end{pmatrix}

Example 2: With any matrix

Let A be the matrix defined by

A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8\\ \end{pmatrix}

The transpose of A will be written

^t A = \begin{pmatrix} 1 & 5\\ 2 & 6 \\ 3 & 7 \\ 4 & 8 \end{pmatrix}

Manufacturing

The transpose has various properties.

Linearity of the transpose

First of all, it is a linear map. It therefore verifies the following property:

\forall A,B \in M_{n,p}(\mathbb{K}) , {}^t (A+B) = {}^t A + {}^t B

As well as this one:

\forall A M_{n,p}(\mathbb{K}), \forall \lambda \in \mathbb K, {}^t (\lambda A) =\lambda {}^t A 

Inverse of the transpose

To calculate its inverse, it's easy, the following formula gives the right result:

({}^tA)^{-1} = {}^t(A^{-1})

Trace of the transpose

For its trace, it's easy, it's the same as the original matrix, so you have to calculate

tr({}^tA ) = tr(A)

Of course, the matrix must be square!

Determinant of the transpose

Same as for the trace, it is equal to the determinant of the original matrix. We therefore have, obviously, the following relation

\det({}^tA) = \det(A)

Product of the transpose

Here, be careful, we reverse. But rest assured, nothing too bad! Here is the formula to remember:

{}^t(AB) = {}^tB {}^tA

It should be noted :

  • A matrix which is equal to its transpose is said to be symmetric.
  • A matrix which is equal to the opposite of its transpose is said to be antisymmetric.

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