Khatri–Rao-szorzat

Sablon:Lektor Mátrixok szorzásánál a Khatri–Rao-szorzat definíciója:

$𝐀\ast 𝐁=({𝐀}_{ij}\otimes {𝐁}_{ij}{)}_{ij}$, ahol az ij indexű blokk m_ip_i × n_jq_j méretű Kronecker-szorzata a megfelelő blokkoknak, feltéve, hogy a két mátrix blokkjainak száma azonos. A szorzat mérete (Σ_i m_ip_i) × (Σ_j n_jq_j).

Példák:

1. példa:

$𝐀=\left[\begin{array}{cc}{𝐀}_{11}& {𝐀}_{12}\\ {𝐀}_{21}& {𝐀}_{22}\end{array}\right]=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right],𝐁=\left[\begin{array}{cc}{𝐁}_{11}& {𝐁}_{12}\\ {𝐁}_{21}& {𝐁}_{22}\end{array}\right]=\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right],$

$𝐀\ast 𝐁=\left[\begin{array}{cc}{𝐀}_{11}\otimes {𝐁}_{11}& {𝐀}_{12}\otimes {𝐁}_{12}\\ {𝐀}_{21}\otimes {𝐁}_{21}& {𝐀}_{22}\otimes {𝐁}_{22}\end{array}\right]=\left[\begin{array}{cccc}1& 2& 12& 21\\ 4& 5& 24& 42\\ 14& 16& 45& 72\\ 21& 24& 54& 81\end{array}\right].$

2. példa: oszloponkénti Khatri–Rao-szorzat:

$𝐂=\left[\begin{array}{ccc}{𝐂}_1& {𝐂}_2& {𝐂}_3\end{array}\right]=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right],𝐃=\left[\begin{array}{ccc}{𝐃}_1& {𝐃}_2& {𝐃}_3\end{array}\right]=\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right],$

kapjuk, hogy:

$𝐂\ast 𝐃=\left[\begin{array}{ccc}{𝐂}_1\otimes {𝐃}_1& {𝐂}_2\otimes {𝐃}_2& {𝐂}_3\otimes {𝐃}_3\end{array}\right]=\left[\begin{array}{ccc}1& 8& 21\\ 2& 10& 24\\ 3& 12& 27\\ 4& 20& 42\\ 8& 25& 48\\ 12& 30& 54\\ 7& 32& 63\\ 14& 40& 72\\ 21& 48& 81\end{array}\right].$

Face-splitting-szorzat

Példák:

$𝐂=\left[\begin{array}{c}{𝐂}_1\\ {𝐂}_2\\ {𝐂}_3\end{array}\right]=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right],𝐃=\left[\begin{array}{c}{𝐃}_1\\ {𝐃}_2\\ {𝐃}_3\end{array}\right]=\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right],$

$𝐂\bullet 𝐃=\left[\begin{array}{c}{𝐂}_1\otimes {𝐃}_1\\ {𝐂}_2\otimes {𝐃}_2\\ {𝐂}_3\otimes {𝐃}_3\end{array}\right]=\left[\begin{array}{ccccccccc}1& 4& 7& 2& 8& 14& 3& 12& 21\\ 8& 20& 32& 10& 25& 40& 12& 30& 48\\ 21& 42& 63& 24& 48& 72& 27& 54& 81\end{array}\right].$

Tulajdonságai

${(𝐀\bullet 𝐁)}^{\mathsf{\text{T}}}={\mathbf{\text{A}}}^{\mathsf{\text{T}}}\ast {𝐁}^{\mathsf{\text{T}}}$

$(𝐀\bullet 𝐁)({𝐀}^{\mathsf{\text{T}}}\ast {𝐁}^{\mathsf{\text{T}}})=\left(𝐀{𝐀}^{\mathsf{\text{T}}}\right)\circ \left(𝐁{𝐁}^{\mathsf{\text{T}}}\right)$,

$(𝐀\ast 𝐁{)}^{\mathsf{\text{T}}}(𝐀\ast 𝐁)=({𝐀}^{\mathsf{\text{T}}}𝐀)\circ ({𝐁}^{\mathsf{\text{T}}}𝐁)$,

$\circ$ - Hadamard-szorzat.

$(𝐀\bullet 𝐁)(𝐂\ast 𝐃)=(𝐀𝐂)\circ (𝐁𝐃)$.

$(𝐀\otimes 𝐁)(𝐂\ast 𝐃)=(𝐀𝐂)\ast (𝐁𝐃)$

$(𝐀\bullet 𝐁)(𝐂\otimes 𝐃)=(𝐀𝐂)\bullet (𝐁𝐃)$

$(𝐀\bullet 𝐋)(𝐁\otimes 𝐌)...(𝐂\otimes 𝐒)(𝐊\ast 𝐓)=(𝐀𝐁...𝐂𝐊)\circ (𝐋𝐌...𝐒𝐓)$

Block Face-Splitting-szorzat

Transzponált Block Face-Splitting-szorzat

Példák:

$𝐀=\left[\begin{array}{cc}{𝐀}_{11}& {𝐀}_{12}\\ {𝐀}_{21}& {𝐀}_{22}\end{array}\right],𝐁=\left[\begin{array}{cc}{𝐁}_{11}& {𝐁}_{12}\\ {𝐁}_{21}& {𝐁}_{22}\end{array}\right],$

$𝐀[\bullet ]𝐁=\left[\begin{array}{cc}{𝐀}_{11}\bullet {𝐁}_{11}& {𝐀}_{12}\bullet {𝐁}_{12}\\ {𝐀}_{21}\bullet {𝐁}_{21}& {𝐀}_{22}\bullet {𝐁}_{22}\end{array}\right]$.

A transzponált Block Face-Splitting-szorzat

$𝐀[\ast ]𝐁=\left[\begin{array}{cc}{𝐀}_{11}\ast {𝐁}_{11}& {𝐀}_{12}\ast {𝐁}_{12}\\ {𝐀}_{21}\ast {𝐁}_{21}& {𝐀}_{22}\ast {𝐁}_{22}\end{array}\right]$.

Tulajdonságai

${(𝐀[\ast ]𝐁)}^{\mathsf{\text{T}}}={\mathbf{\text{A}}}^{\mathsf{\text{T}}}[\bullet ]{𝐁}^{\mathsf{\text{T}}}$

Irodalom

• Sablon:Cite journal
• Sablon:Citation
• Matrix Algebra & Its Applications to Statistics & Econometrics./C. R. Rao with M. Bhaskara Rao. - World Scientific. - 1998. - P. 216.