Rayleigh–Ritz method: Difference between revisions

Revision as of 18:51, 23 March 2022

Revision as of 21:33, 23 March 2022

Line 116:

with

:<math> \mathbf {U} = \begin{bmatrix}

0 & -1\\

1 & 0\\

0 & 0\\

Line 129:

\quad

\mathbf {V}_h = \begin{bmatrix}

- \sqrt{2}/2 & \sqrt{2}/2\\

- \sqrt{2} & -\sqrt{2}

\end{bmatrix}.

</math>

Thus we already obtain the singular values 2 and 1 from <math>\Sigma</math> and from <math>\mathbf {U}</math> the corresponding two left singular vectors <math>u</math> as <math>[0, 1, 0, 0, 0]^*</math> and <math>[-1, 0, 0, 0, 0]^*</math>, which span the column-space of the matrix <math>W</math>, explaining why the approximations are exact for the given <math>W</math>.

Finally, step 3 computes the matrix <math>V_h = \mathbf {V}_h W^*</math>

: <math>

\mathbf {V}_h = \begin{bmatrix}

- \sqrt{2}/2 & \sqrt{2}/2\\

- \sqrt{2} & -\sqrt{2}

\end{bmatrix}

\,

= \begin{bmatrix}

- \sqrt{2}/2 & \sqrt{2}/2 & 0 & 0\\

- \sqrt{2} & -\sqrt{2} & 0 & 0

\end{bmatrix} =

\begin{bmatrix}

0 & -1 & 0 & 0\\

-1 & 0 & 0 & 0

\end{bmatrix}

</math>

recovering from its rows the two right singular vectors <math>v</math> as <math>[0, -1, 0, 0]^*</math> and <math>[-1, 0, 0, 0]^*</math>.

We validate <math>Mv=\sigma u</math>

: <math>

\begin{bmatrix}

1 & 0 & 0 & 0\\

0 & 2 & 0 & 0\\

0 & 0 & 3 & 0\\

0 & 0 & 0 & 4\\

0 & 0 & 0 & 0

\end{bmatrix}

\,

\begin{bmatrix}0\\ 1\\ 0\\ 0\end{bmatrix}

= \, 2 \,

\begin{bmatrix}0\\ 1\\ 0\\ 0\\ 0\end{bmatrix}

</math>

and <math>M^* u=\sigma v</math>

: <math>

\begin{bmatrix}

1 & 0 & 0 & 0 & 0\\

0 & 2 & 0 & 0 & 0\\

0 & 0 & 3 & 0 & 0\\

0 & 0 & 0 & 4 & 0

\end{bmatrix}

\,

\begin{bmatrix}0\\ 1\\ 0\\ 0\\ 0\end{bmatrix}

= \, 2 \,

\begin{bmatrix}0\\ 1\\ 0\\ 0\end{bmatrix}.

</math>

== Derivation from calculus of variations ==