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Exact SVD algorithm of matrix with symbolic entries
I am making a library with symbolic computations which supports matrices. A matrix may have symbolic entries (eg be over the ring of polynomials of a variable x ie $\mathbb{Q}[x]$).
I have implemented almost all useful linear algebra algorithms and decompositions using exact/symbolic computations (and in some cases fraction-free algorithms) (eg REF, RREF, SNF, INVERSE, PSEUDO-INVERSE, LU, QR, RANK factorisations/decompositions, ROWSPACE, COLUMNSPACE, NULLSPACE etc). The only needed algorithm which I cannot do with exact/symbolic computations is SVD/EVD. I only find numerical algorithms which solve the problem numericaly / approximately and employing "irrational" computations (eg square roots) which are not exact (note: I dont mind if an exact/symbolic algorithm employs square roots, since I can handle these symbolicaly if needed without actually computing square roots).Any exact/symbolic algorithm for SVD/EVD or any way to compute SVD using one of the decompositions I already have and which are exact?

For n odd and positive integer, proof that $\frac{\sin <!--\; \; -->(nx)}{\sin <!--\; \; -->(x)}=(-<!--\; -\; -->4{)}^{(n-<!--\; -\; -->1)/2}\underset{1\le <!--\; \le \; -->j\le <!--\; \le \; -->(n-<!--\; -\; -->1)/2}{\prod <!--\; \prod \; -->}({\sin }^2<!--\; \; -->(x)-<!--\; -\; -->{\sin }^2<!--\; \; -->(\frac{2\mathrm{\pi <!--\; \pi \; -->}j}{n}))$
It says this is elementary, and suggest first proving that $\frac{\sin <!--\; \; -->(nx)}{\sin <!--\; \; -->(x)}$ is a polynomial of degree $(n-<!--\; -\; -->1)/2$ in ${\sin }^2<!--\; \; -->(x)$, then remark that it has ${\sin }^2<!--\; \; -->(\frac{2\mathrm{\pi <!--\; \pi \; -->}j}{n})$ as roots, then comparing coefficients of $e^{i(n-<!--\; -\; -->1)/2}$ to get $(-<!--\; -\; -->4{)}^{(n-<!--\; -\; -->1)/2}$.

How can I compute the average eigenvalue of a parametrised matrix?
I have a family of $n\times <!--\; \times \; -->n$ parameterised Hermitian matrices, the simplest of which is of the form:
$$\mathrm{M}(t)=f(t)\mathrm{I}+\mathrm{Q}$$
where f(t) is a polynomial in t, I is the usual $n\times <!--\; \times \; -->n$ identity matrix and Q is a known $n\times <!--\; \times \; -->n$ Hermitian matrix. I need to compute the (arithmetic) mean of each of the eigenvalues $\{{\mathrm{\lambda <!--\; \lambda \; -->}}_{\mathrm{\alpha <!--\; \alpha \; -->}}(t)\}$ of M(t) over an interval $a\le <!--\; \le \; -->t\le <!--\; \le \; -->b$. What is the best way to do this?

Find a polynomial f(x) that satisfies this equation:
$x{f}^{″}(x)+f(x)=x^2$

Find a polynomial P(x) with real coefficients having a degree 6, leading coefficient 5, and zeros 7, 0 (multiplicity 3) and 2 - 3i

The polynomial $x^2+3x+2$ has only $3$ zeros in $Z6$

Find the quotient and remainder when is divided by in

Factor completly.
$8(2x+3{)}^2(x-<!--\; -\; -->7{)}^4-<!--\; -\; -->(2x+3{)}^3(x-<!--\; -\; -->7{)}^3$

Evaluate $$(6+13)\times <!--\; \times \; -->13-<!--\; -\; -->11\times <!--\; \times \; -->7+8$$.

Perform the operations in the correct order: $$8\times <!--\; \times \; -->(7-<!--\; -\; -->8)\times <!--\; \times \; -->8+11\times <!--\; \times \; -->13$$.

Compute $$7-<!--\; -\; -->14+(14+5)\times <!--\; \times \; -->12÷<!--\; ÷\; -->6$$.

Compute $$8+14\times <!--\; \times \; -->(11-<!--\; -\; -->6)\times <!--\; \times \; -->6\times <!--\; \times \; -->8$$ using the correct order of operations.

What is negative twelve times negative one hundred ninety-seven?

What is the product of -109 and 76?

What is negative three hundred sixty divided by negative thirty?

Compute -39(-183).

What is -139 times -12?

Solve 21-?=10.

Compute 988+384.