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Results tagged with linear-algebra
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user 140320
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
1
vote
Condition of tri-diagonal matrix A such that UAU^t=A for all orthogonal matrix U
In general, this doesn't hold for diagonal matrices $A$ in general.
Consider
$$A=\begin{bmatrix} 1 & 0 &0 \\0 & 2 &0 \\ 0& 0& 3 \end{bmatrix}$$
and
$$U = \begin{bmatrix} 0 &0 & 1 \\1 & 0 &0 \\ 0& 1& …
1
vote
Is the solution of this linear system always positive definite?
I think the answer is yes. Write $$P^\mathrm{t}(J_n \circ PSP^\mathrm{t})P=P^\mathrm{t}(PSP^\mathrm{t}-I_n\circ PSP^\mathrm{t})P=S-P^\mathrm{t}(I_n \circ PSP^\mathrm{t})P$$
so the solution satisfies
$ …