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Results tagged with matrix-theory
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user 9091
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
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Determine unknown matrix function of particular form from known points
We have a black box $f:X\mapsto B^{-1}(X-A)(DB^{-1}(X-A)+C)^{-1}$.
We are looking for approximations of $A,B,C,D\in M_n$, where $A,B,C$ are invertible.
Note that if $(A,B,C,D)$ is a solution, then $(A …
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Solving linear matrix equation
I feel like you are drowning in a glass of water.
Putting $AC=U\in M_2$ and $BC=V\in M_2$, we obtain
(*) $UXU^T-VXV^T=L$ in $M_2$.
If $X$ is a symmetric solution of (*), then $L$ too.
For generic $U,V …
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22
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What is the time complexity of truncated SVD?
@ user40484 , fortunately your estimate for the complexity of SVD is not optimal. Otherwise, you put unemployed specialists in image compression. The complexity is in $O(\min(mn^2,m^2n))$.
Assume th …
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4
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Calculating the dimension of the algebra generated by some given matrices
Your $X_1,X_2$ have a common invariant subspace $span([1,0]^T)$. If $K$ is algebraically closed and if $X_1,X_2$ have no non-trivial common invariant subspaces (in particular if $X_1,X_2$ are randomly …
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Perturbation of Cholesky decomposition for matrix inversion
Let $A=LL^T,\lambda=\mu^2,f:X\rightarrow X^{-1}$. That follows is an approximate approach in $O(n^2)$ that is valid only if $\lambda$ is small with respect to $\inf(spectrum(A))$. $Df_A(H)=-A^{-1}HA^{ …
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