Formula for Matrix Power Form
📂Matrix Algebra Formula for Matrix Power Form For the matrix X = [ x i j ] ∈ R n × n X = [x_{ij}] \in \mathbb{R}^{n \times n} X = [ x ij ] ∈ R n × n , the following holds.
[ X X ] i j = ∑ k = 1 n x i k x k j
[XX]_{ij} = \sum_{k=1}^{n} x_{ik} x_{kj}
[ XX ] ij = k = 1 ∑ n x ik x kj
X X = X 2 = [ ∑ k = 1 n x 1 k x k 1 ⋯ ∑ k = 1 n x 1 k x k n ⋮ ⋱ ⋮ ∑ k = 1 n x n k x k 1 ⋯ ∑ k = 1 n x n k x k n ]
XX = X^{2} = \begin{bmatrix}
\sum\limits_{k=1}^{n} x_{1k} x_{k1} & \cdots & \sum\limits_{k=1}^{n} x_{1k} x_{kn} \\ \vdots & \ddots & \vdots \\ \sum\limits_{k=1}^{n} x_{nk} x_{k1} & \cdots & \sum\limits_{k=1}^{n} x_{nk} x_{kn}
\end{bmatrix}
XX = X 2 = k = 1 ∑ n x 1 k x k 1 ⋮ k = 1 ∑ n x nk x k 1 ⋯ ⋱ ⋯ k = 1 ∑ n x 1 k x kn ⋮ k = 1 ∑ n x nk x kn
If X X X is a symmetric matrix ,
X 2 = [ ∑ k = 1 n ( x 1 k ) 2 ⋯ ∑ k = 1 n x 1 k x k n ⋮ ⋱ ⋮ ∑ k = 1 n x n k x k 1 ⋯ ∑ k = 1 n ( x n k ) 2 ]
X^{2} = \begin{bmatrix}
\sum\limits_{k=1}^{n} (x_{1k})^{2} & \cdots & \sum\limits_{k=1}^{n} x_{1k} x_{kn} \\ \vdots & \ddots & \vdots \\ \sum\limits_{k=1}^{n} x_{nk} x_{k1} & \cdots & \sum\limits_{k=1}^{n} (x_{nk})^{2}
\end{bmatrix}
X 2 = k = 1 ∑ n ( x 1 k ) 2 ⋮ k = 1 ∑ n x nk x k 1 ⋯ ⋱ ⋯ k = 1 ∑ n x 1 k x kn ⋮ k = 1 ∑ n ( x nk ) 2
Generalization The following holds.
X X X = X 3 = [ ∑ k , ℓ = 1 n x 1 k x k ℓ x ℓ 1 ⋯ ∑ k , ℓ = 1 n x 1 k x k ℓ x ℓ n ⋮ ⋱ ⋮ ∑ k , ℓ = 1 n x n k x k ℓ x ℓ 1 ⋯ ∑ k , ℓ = 1 n x n k x k ℓ x ℓ n ]
XXX = X^{3} = \begin{bmatrix}
\sum\limits_{k,\ell=1}^{n}x_{1k}x_{k\ell}x_{\ell 1} & \cdots & \sum\limits_{k,\ell=1}^{n}x_{1k}x_{k\ell}x_{\ell n} \\ \vdots & \ddots & \vdots \\ \sum\limits_{k,\ell=1}^{n}x_{nk}x_{k\ell}x_{\ell 1} & \cdots & \sum\limits_{k,\ell=1}^{n}x_{nk}x_{k\ell}x_{\ell n}
\end{bmatrix}
XXX = X 3 = k , ℓ = 1 ∑ n x 1 k x k ℓ x ℓ 1 ⋮ k , ℓ = 1 ∑ n x nk x k ℓ x ℓ 1 ⋯ ⋱ ⋯ k , ℓ = 1 ∑ n x 1 k x k ℓ x ℓ n ⋮ k , ℓ = 1 ∑ n x nk x k ℓ x ℓ n
Regarding the set K = { k 1 , k 2 , … , k ∣ K ∣ } K = \left\{ k_{1}, k_{2}, \dots, k_{|K|} \right\} K = { k 1 , k 2 , … , k ∣ K ∣ } , the following holds.
X ∣ K ∣ = [ ∑ K x 1 k 1 x k 1 k 2 ⋯ x k ∣ K ∣ 1 ⋯ ∑ K x 1 k 1 x k 1 k 2 ⋯ x k ∣ K ∣ n ⋮ ⋱ ⋮ ∑ K x n k 1 x k 1 k 2 ⋯ x k ∣ K ∣ 1 ⋯ ∑ K x n k 1 x k 1 k 2 ⋯ x k ∣ K ∣ n ]
X^{|K|} = \begin{bmatrix}
\sum\limits_{K} x_{1k_{1}}x_{k_{1}k_{2}}\cdots x_{k_{|K|}1} & \cdots & \sum\limits_{K} x_{1k_{1}}x_{k_{1}k_{2}}\cdots x_{k_{|K|}n} \\ \vdots & \ddots & \vdots \\ \sum\limits_{K} x_{nk_{1}}x_{k_{1}k_{2}}\cdots x_{k_{|K|}1} & \cdots & \sum\limits_{K} x_{nk_{1}}x_{k_{1}k_{2}}\cdots x_{k_{|K|}n}
\end{bmatrix}
X ∣ K ∣ = K ∑ x 1 k 1 x k 1 k 2 ⋯ x k ∣ K ∣ 1 ⋮ K ∑ x n k 1 x k 1 k 2 ⋯ x k ∣ K ∣ 1 ⋯ ⋱ ⋯ K ∑ x 1 k 1 x k 1 k 2 ⋯ x k ∣ K ∣ n ⋮ K ∑ x n k 1 x k 1 k 2 ⋯ x k ∣ K ∣ n
Explanation The formula is valid not only for powers of a single matrix but also for products of multiple matrices. Additionally, if a transpose is included, the order of the indices of the matrix should be reversed. For example, for the n × n n \times n n × n matrix A , B , C A, B, C A , B , C ,
[ A B ] i j = [ ∑ k a i k b k j ] [ A B T ] i j = [ ∑ k a i k b j k ] [ A B C ] i j = [ ∑ k , s a i k b k s c s j ] [ A B T C ] i j = [ ∑ k , s a i k b s k c s j ]
\begin{align*}
[AB]_{ij} &= \left[ \sum_{k} a_{ik}b_{kj}\right] \\[1em]
[AB^{\mathsf{T}}]_{ij} &= \left[ \sum_{k} a_{ik}b_{jk}\right] \\[1em]
[ABC]_{ij} &= \left[ \sum_{k,s} a_{ik}b_{ks}c_{sj}\right] \\[1em]
[AB^{\mathsf{T}}C]_{ij} &= \left[ \sum_{k,s} a_{ik}b_{sk}c_{sj}\right]
\end{align*}
[ A B ] ij [ A B T ] ij [ A BC ] ij [ A B T C ] ij = [ k ∑ a ik b kj ] = [ k ∑ a ik b jk ] = k , s ∑ a ik b k s c s j = k , s ∑ a ik b s k c s j
Proof It can be derived through simple calculations. The same can be verified for the power form as well.
X X = [ x 11 x 12 ⋯ x 1 n x 21 x 22 ⋯ x 2 n ⋮ ⋮ ⋱ ⋮ x n 1 x n 2 ⋯ x n n ] [ x 11 x 12 ⋯ x 1 n x 21 x 22 ⋯ x 2 n ⋮ ⋮ ⋱ ⋮ x n 1 x n 2 ⋯ x n n ]
XX
= \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n} \\ x_{21} & x_{22} & \cdots & x_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & \cdots & x_{nn} \end{bmatrix} \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n} \\ x_{21} & x_{22} & \cdots & x_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & \cdots & x_{nn} \end{bmatrix}
XX = x 11 x 21 ⋮ x n 1 x 12 x 22 ⋮ x n 2 ⋯ ⋯ ⋱ ⋯ x 1 n x 2 n ⋮ x nn x 11 x 21 ⋮ x n 1 x 12 x 22 ⋮ x n 2 ⋯ ⋯ ⋱ ⋯ x 1 n x 2 n ⋮ x nn
⟹ [ X X ] 11 = x 11 x 11 + x 12 x 21 + ⋯ + x 1 n x n 1 = ∑ k x 1 k x k 1 [ X X ] 12 = x 11 x 12 + x 12 x 22 + ⋯ + x 1 n x n 2 = ∑ k x 1 k x k 2 ⋮ [ X X ] n n = x n 1 x 1 n + x n 2 x 2 n + ⋯ + x n n x n n = ∑ k x n k x k n
\implies
\begin{align*}
[XX]_{11} &= x_{11} x_{11} + x_{12} x_{21} + \cdots + x_{1n} x_{n1} = \sum\limits_{k} x_{1k} x_{k1} \\
[XX]_{12} &= x_{11} x_{12} + x_{12} x_{22} + \cdots + x_{1n} x_{n2} = \sum\limits_{k} x_{1k} x_{k2} \\
& \vdots \\
[XX]_{nn} &= x_{n1} x_{1n} + x_{n2} x_{2n} + \cdots + x_{nn} x_{nn} = \sum\limits_{k} x_{nk} x_{kn}
\end{align*}
⟹ [ XX ] 11 [ XX ] 12 [ XX ] nn = x 11 x 11 + x 12 x 21 + ⋯ + x 1 n x n 1 = k ∑ x 1 k x k 1 = x 11 x 12 + x 12 x 22 + ⋯ + x 1 n x n 2 = k ∑ x 1 k x k 2 ⋮ = x n 1 x 1 n + x n 2 x 2 n + ⋯ + x nn x nn = k ∑ x nk x kn
⟹ X X = [ ∑ k x 1 k x k 1 ⋯ ∑ k x 1 k x k n ⋮ ⋱ ⋮ ∑ k x n k x k 1 ⋯ ∑ k x n k x k n ]
\implies
XX = \begin{bmatrix}
\sum\limits_{k} x_{1k} x_{k1} & \cdots & \sum\limits_{k} x_{1k} x_{kn} \\ \vdots & \ddots & \vdots \\ \sum\limits_{k} x_{nk} x_{k1} & \cdots & \sum\limits_{k} x_{nk} x_{kn}
\end{bmatrix}
⟹ XX = k ∑ x 1 k x k 1 ⋮ k ∑ x nk x k 1 ⋯ ⋱ ⋯ k ∑ x 1 k x kn ⋮ k ∑ x nk x kn
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