The Matrix Transpose {矩阵转置}
- [1. Definition: Matrix Transpose](#1. Definition: Matrix Transpose)
- [2. Properties of the Matrix Transpose](#2. Properties of the Matrix Transpose)
- References
1. Definition: Matrix Transpose
Let A \mathbf{A} A be an m × n m\times n m×n matrix. The tranpsose of A \mathbf{A} A, denoted A T \mathbf {A}^{\mathrm {T}} AT, is the n × m n\times m n×m matrix whose columns are the respective rows of A \mathbf{A} A.
The transpose of a matrix is an operator that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the matrix A \mathbf{A} A to produce another matrix, often denoted A T \mathbf {A}^{\mathrm {T}} AT.
transposition [ˌtrænspə'zɪʃ(ə)n]
n. 换位；移项；词序改变；变调 (曲)The transpose A T \mathbf {A}^{\mathrm {T}} AT of a matrix A \mathbf{A} A can be obtained by reflecting the elements along its main diagonal.
The i i ith row, j j jth column element of A T \mathbf {A}^{\mathrm {T}} AT is the j j jth row, i i ith column element of A \mathbf{A} A:
A T \] i j = \[ A \] j i . \\left\[ \\mathbf{A}\^\\mathrm {T} \\right\]_{ij} = \\left\[ \\mathbf{A} \\right\]_{ji}. \[AT\]ij=\[A\]ji. 注意： A T \\mathbf {A}\^{\\mathrm {T}} AT (转置矩阵) 与 A − 1 \\mathbf {A} \^{-1} A−1 (逆矩阵) 不同。 * Example A = \[ 1 2 3 4 5 6 \] \\mathbf {A} = \\left\[ \\begin{array}{ccc}{1}\&{2}\&{3}\\\\{4}\&{5}\&{6}\\end{array} \\right\] A=\[142536
Note that A \mathbf{A} A is a 2 × 3 2 \times 3 2×3 matrix, so A T \mathbf {A}^{\mathrm {T}} AT will be a 3 × 2 3 \times 2 3×2 matrix. By the definition, the first column of A T \mathbf {A}^{\mathrm {T}} AT is the first row of A \mathbf{A} A; the second column of A T \mathbf {A}^{\mathrm {T}} AT is the second row of A \mathbf{A} A. Therefore,
A T = [ 1 4 2 5 3 6 ] . \mathbf {A}^{\mathrm {T}} = \left[\begin{array}{cc}{1}&{4}\\{2}&{5}\\{3}&{6}\end{array}\right]. AT= 123456 .
2. Properties of the Matrix Transpose
Let A \mathbf {A} A and B \mathbf {B} B be matrices and k k k be a scalar.
- ( A + B ) T = A T + B T (\mathbf {A}+\mathbf {B})^{\mathrm {T}}=\mathbf {A}^{\mathrm {T}}+\mathbf {B}^{\mathrm {T}} (A+B)T=AT+BT and ( A − B ) T = A T − B T (\mathbf {A}-\mathbf {B})^{\mathrm {T}}=\mathbf {A}^{\mathrm {T}}-\mathbf {B}^{\mathrm {T}} (A−B)T=AT−BT
- ( k A ) T = k A T (k\mathbf {A})^{\mathrm {T}}=k\mathbf {A}^{\mathrm {T}} (kA)T=kAT
The transpose of a scalar is the same scalar.
标量的转置是同样的标量。
- ( A B ) T = B T A T (\mathbf {AB})^{\mathrm {T}}=\mathbf {B}^{\mathrm {T}}\mathbf {A}^{\mathrm {T}} (AB)T=BTAT
By induction, this result extends to the general case of multiple matrices, so ( A 1 A 2 . . . A k − 1 A k ) T = A k T A k − 1 T ... A 2 T A 1 T (\mathbf {A}{1}\mathbf {A}{2}...\mathbf {A}{k-1}\mathbf {A}{k})^{\mathrm {T}} = \mathbf {A}{k}^{\mathrm {T}}\mathbf {A}{k-1}^{\mathrm {T}}...\mathbf {A}{2}^{\mathrm {T}}\mathbf {A}{1}^{\mathrm {T}} (A1A2...Ak−1Ak)T=AkTAk−1T...A2TA1T.
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( A − 1 ) T = ( A T ) − 1 (\mathbf {A}^{-1})^{\mathrm {T}}=(\mathbf {A}^{\mathrm {T}})^{-1} (A−1)T=(AT)−1
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( A T ) T = A (\mathbf {A}^{\mathrm {T}})^{\mathrm {T}}=\mathbf {A} (AT)T=A
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A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A \mathbf {A} A is symmetric if A T = A \mathbf {A} ^{\text{T}}=\mathbf {A} AT=A.
A matrix A \mathbf {A} A is symmetric if A T = A \mathbf {A} ^{\text{T}}=\mathbf {A} AT=A.
一个矩阵的转置等于自身的方块矩阵叫做对称矩阵。
symmetric [sɪ'metrɪk]
adj. 对称的- A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A \mathbf {A} A is skew-symmetric if A T = − A \mathbf {A} ^{\text{T}}=-\mathbf {A} AT=−A.
A matrix A \mathbf {A} A is skew symmetric if A T = − A \mathbf {A} ^{\text{T}}=-\mathbf {A} AT=−A.
一个矩阵的转置等于它的负矩阵的方块矩阵叫做斜对称矩阵。
skew [skjuː]
n. 斜交；扭曲；斜砌石；歪轮
adj. 歪的；弯曲的；误用的；曲解的
v. 歪曲；曲解；使不公允；影响 ... 的准确性- Given any matrix A \mathbf {A} A, the matrices A A T \mathbf {A}\mathbf {A}^{T} AAT and A T A \mathbf {A}^{T}\mathbf {A} ATA are symmetric.
If A \mathbf {A} A is an m × n m\times n m×n matrix and A T \mathbf {A}^{\mathrm {T}} AT is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T \mathbf {A}\mathbf {A}^{T} AAT is m × m m\times m m×m and A T A \mathbf {A}^{T}\mathbf {A} ATA is n × n n\times n n×n. Furthermore, these products are symmetric matrices.
Indeed, the matrix product A A T \mathbf {A}\mathbf {A}^{T} AAT has entries that are the inner product of a row of A \mathbf {A} A with a column of A T \mathbf {A}^{\mathrm {T}} AT. But the columns of A T \mathbf {A}^{\mathrm {T}} AT are the rows of A \mathbf {A} A, so the entry corresponds to the inner product of two rows of A \mathbf {A} A. If p i j p_{ij} pij is the entry of the product ( A A T \mathbf {A}\mathbf {A}^{T} AAT), it is obtained from i i i row of A \mathbf {A} A with j j j column of A T \mathbf {A}^{\mathrm {T}} AT. The entry p j i p_{ji} pji is obtained from j j j row of A \mathbf {A} A with i i i column of A T \mathbf {A}^{\mathrm {T}} AT, thus p i j = p j i p_{ij} = p_{ji} pij=pji, and the product matrix is symmetric. Similarly, the product A T A \mathbf {A}^{T}\mathbf {A} ATA is a symmetric matrix.
A quick proof of the symmetry of A A T \mathbf {A}\mathbf {A}^{T} AAT results from the fact that it is its own transpose:
( A A T ) T = ( A T ) T A T = A A T \left(\mathbf{A} \mathbf{A}^\text{T}\right)^\text{T} = \left(\mathbf{A}^\text{T}\right)^\text{T} \mathbf{A}^\text{T}= \mathbf{A} \mathbf{A}^\text{T} (AAT)T=(AT)TAT=AAT.
References
1\] Yongqiang Cheng (程永强),