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Wize University Linear Algebra Textbook > Matrices

Matrix Transpose

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Matrix Transpose

The transpose of matrix AA, denoted by ATA^T, is obtained by changing its columns into rows (and vice-versa).
Example
A=[abcdefghi],AT=[adgbehcfi]A= \begin{bmatrix} \colorOne{a}&\colorOne{b}&\colorOne{c}\\[0.5em] \colorTwo{d}&\colorTwo{e}&\colorTwo{f}\\[0.5em] g&h&i \end{bmatrix}, \quad A^T=\begin{bmatrix} \colorOne{a}&\colorTwo{d}&g\\[0.5em] \colorOne{b}&\colorTwo{e}&h\\[0.5em] \colorOne{c}&\colorTwo{f}&i \end{bmatrix}
In general: (aT)ij=aji(a^T)_{ij}=a_{ji}
In other words, entry (i,j)(i,j) in AA will become entry (j,i)(j,i) in ATA^T
Wize Concept
If AA is of size m×nm\times n, then ATA^T will be of size n×mn\times m (swap the dimensions!)

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Properties of the Transpose

Let AA and BB be matrices, and let k∈Rk \in \reals.
  • (AT)T=A(A^T)^T=A
  • (A+B)T=AT+BT(A+B)^T=A^T+B^T
  • (AB)T=BTAT(AB)^T=B^TA^T
  • (kA)T=k(AT)(kA)^T=k(A^T)
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Symmetric Matrices

A matrix AA is symmetric if AT=AA^T = A.
That is, a symmetric matrix is equal to its own transpose.
Wize Tip
Only square matrices can be symmetric.
Think of the main diagonal as the line of symmetry.
Example
The matrix MM is symmetric:
M=[379725950],MT=[379725950]M= \left[\begin{array}{rrr} \colorOne{3}&\colorOne{7}&\colorOne{9}\\[0.5em] \colorTwo{7}&\colorTwo{2}&\colorTwo{5}\\[0.5em] 9&5&0\\ \end{array} \right] , \quad M^T = \left[ \begin{array}{rrr} \colorOne{3}&\colorTwo{7}&9\\[0.5em] \colorOne{7}&\colorTwo{2}&5\\[0.5em] \colorOne{9}&\colorTwo{5}&0\\ \end{array} \right]
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Skew-Symmetric Matrix

A matrix is skew-symmetric if AT=−AA^T=-A (the transpose is equal to the negative of the matrix).

Wize Tip
Only square matrices can be skew-symmetric.
Every entry on the main diagonal must be 0.
Example
The matrix PP is skew-symmetric:
P=[01−5−10−115110]P=\left[\begin{array}{rrr} 0&1&-5\\[0.5em] -1&0&-11\\[0.5em] 5&11&0\\ \end{array}\right]
Notice that PT=−PP^T=-P:
PT=[01−5−10−115110]T=[0−151011−5−110]=(−1)⋅[01−5−10−115110]=−PP^T=\left[\begin{array}{rrr} \colorOne{0}&\colorOne{1}&\colorOne{-5}\\[0.5em] \colorTwo{-1}&\colorTwo{0}&\colorTwo{-11}\\[0.5em] 5&11&0\\ \end{array}\right]^T = \left[\begin{array}{rrr} \colorOne{0}&\colorTwo{-1}&5\\[0.5em] \colorOne{1}&\colorTwo{0}&11\\[0.5em] \colorOne{-5}&\colorTwo{-11}&0\\ \end{array}\right] =(-1)\cdot\left[\begin{array}{rrr} 0&1&-5\\[0.5em] -1&0&-11\\[0.5em] 5&11&0\\ \end{array}\right] =-P
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Example: Matrix Transpose

Transpose the matrix A=[135246]A=\left[\begin{array}{rrr}1 & 3 & 5\\2 & 4 &6\end{array}\right], and find the product AATAA^T.
Determine whether AATAA^T is symmetric, skew-symmetric, or neither.
AT=[123456]A^T=\left[\begin{array}{rr} 1 & 2\\3&4\\5 & 6\end{array}\right]
AAT=[135246][123456]=[35444456]AA^T = \left[\begin{array}{rrr}1 & 3 & 5\\2 & 4 &6\end{array}\right] \left[\begin{array}{rr}1 & 2\\3&4\\5&6\\\end{array}\right] = \left[\begin{array}{rr}35 & 44\\44&56\end{array}\right]
Since this matrix is symmetric across the main diagonal, and since the transpose is equal to itself, AATAA^T is symmetric.
Note: (AAT)T=(AT)T(A)T=AAT(\colorOne{AA^T})^T = (A^T)^T(A)^T=\colorOne{AA^T}, so given any matrix AA, AATAA^T is always symmetric.
checklist
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Prove that for any square matrix AA, the matrix A−ATA-A^T is skew-symmetric.
Given C=[431]C = \left[ \begin{array}{rrr} 4 & 3 & 1 \end{array} \right] and D=[131245]D = \left[ \begin{array}{rrr} 1 & 3 & 1\\ 2 & 4 & 5\\ \end{array} \right], find the following entries.
Extra Practice
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 25
Show that A‾T  ⁣A‾ \bcb{\boldsymbol{ \mtran{\A}\ul{A}}} \, is a symmetric matrix for any n×n\bcb{\boldsymbol{ n \times n}} matrix   ⁣A‾\bcb{\boldsymbol{ \ul{A} }}.
Practice: Matrix Operations
Given that A=[−123−4]A=\begin{bmatrix} -1&2\\3&-4 \end{bmatrix} and B=[11−22]B=\begin{bmatrix}1&1\\-2&2\end{bmatrix}, let C=(3A−2BT+I2)TC=(3A-2B^T+I_2)^T. Determine C22C_{22}.
Transpose of a Matrix
Prove that, for an invertible matrix A‾\bcb{\A} , A‾T\bcb{\mtran{\A}} is also invertible, and (AT)−1=(A−1)T\bcb{(A^T )^ {−1} = (A^{−1} )^ T}.
Practice Question 8: Matrix Transpose
Practice Question: Matrix Transpose
Given the following matrices, find the values of a,ba, b and cc such that −3A=BT-3A=B^T.
A=[13−60152]A=\begin{bmatrix} 1&3&-6\\0&15&2 \end{bmatrix} and B=[−3a−95b2c−6]B=\begin{bmatrix}-3&a\\-9&5b\\2c&-6\end{bmatrix}
Matrix Transpose
Prove that for any square matrix AA, the matrix A−ATA-A^T is skew-symmetric.
Example: Matrix Operations
Example:
Given the matrices 𝐴=[100−3−22]𝐴=\begin{bmatrix}1&0\\0&-3\\-2&2\end{bmatrix}, B=[30201−2]\text{B}=\begin{bmatrix}3&0\\2&0\\1&-2\end{bmatrix}, and C=[12010−1]C=\begin{bmatrix}1&2&0\\1&0&-1\end{bmatrix},
a) Find 2A−B2A-B.
Example: Matrix Transpose
Example:
Find the transpose of 𝐴=[1−3−2405]𝐴=\begin{bmatrix}1&-3\\-2&4\\0&5\end{bmatrix}, then state the sizes of AA and ATA^T.
Matrix Multiplication and Transpose
Suppose that A=[2−11103]A=\begin{bmatrix} 2&-1\\ 1&1\\ 0&3 \end{bmatrix} and B=[03−2−110]B=\begin{bmatrix} 0&3\\ -2&-1\\ 1&0 \end{bmatrix}.
Find 2ATB−(BTA)T−ATIB2A^TB-(B^TA)^T-A^TIB, where II is the 3x3 identity matrix.