# Questions tagged [matrix-decomposition]

Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.

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### Why $L^{-1}U^T=D$ in LU decomposition?

I learned that given a matrix $A$, we can apply LU decomposition to get $A=LU$, where $L$ is lower triangular and $U$ is upper triangular. Further, if $A$ is symmetric (or Hermitian for complex $A$), ...
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### Growth factor of specific matrix

Recalling that $$g_n(A) = \frac{||U||_{max}}{||A||_{max}}$$ where $A=LU$ with $U$ being upper triangular and $L$ unit lower triangular, and $|| \cdot ||_{max}$ is the largest element of $(|a_{ij}|)$. ...
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### Determining a basis for the column space of A from its LU factorisation?

There are two ways of finding a basis for the column space of A from A = LU. The pivot columns in U correspond to the linearly independent columns of A, which in turn form a basis for the column ...