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Could someone help me better understand algorithm 23.1 of “Numerical Linear Algebra” (by Lloyd Threfethen). You can see it here .

What’s being calculated at each step of the outer loop? Given that $A_{k-1}=R_k^*A_kR_k$, I suppose that the answer is $A_k^*R_k$. Am I wrong?

- Find the eigenvalues of a matrix with ones in the diagonal, and all the other elements equal
- Proving every real symmetric matrix is congruent to the canonical form $\mathrm{diag}(\mathbf I,\mathbf{-I},\mathbf0)$
- $SL(n,\mathbb R)$ diffeomorphic to $SO(n) \times \mathbb R^{n(n+1)/2-1}$?
- Is Singular value decomposition suitable for solving matrix equations Ax=b?
- A normal matrix with real eigenvalues is Hermitian
- Polar decomposition of real matrices

On the same page it is also stated that the computational cost can be evaluated as follows:

$\Sigma_{k=1}^m\Sigma_{j=k+1}^m2(m-j)\approx2\Sigma_{k=1}^m\Sigma_{j=1}^kj\approx\Sigma_{k=1}^mk^2\approx(1/3)m^3 flops$

I understand the last two approximations, but what about the first one?

Edit: algorithm 23.1:

```
R = A
for k = 1 to m
for j = k+1 to m
R_{j,j:m} = R_{j,j:m}-R_{k,j:m}(conj(R_{kj})/R_{kk});
R_{k,k:m}=R_{k,k:m}/R_{k,k};
```

- On monomial matrix (generalized permutation matrix )
- Prove that matrix is symmetric and positive definite given the fact that $A+iB$ is.
- Singular values of $AB$ and $BA$ matrices
- Direct relation between similar matrices
- Set invariant under group action
- Matrices that Differ only in Diagonal of Decomposition
- Matrix exponential of a skew symmetric matrix
- QR factorization of a special structured matrix

The algorithm seems to be for the LDLT decomposition, the Cholesky decomposition would involve the square roots of the diagonal elements.

And

\begin{align}

\sum_{k=1}^m\sum_{j=k+1}^m2(m-j)

&=2\sum_{1\le k<j\le m}^m(m-j)

=2\sum_{j=1}^m(j-1)(m-j)\\

&=2m(0+1+2+…+(m-1))-2(0⋅1+1⋅2+2⋅3+3⋅4+…+m⋅(m-1))\\

&=m^2(m-1)-\tfrac23(m-1)m(m+1)=\tfrac13(m-2)(m-1)m

\end{align}

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