  We study the left-right action of $\operatorname{SL}_n \times
\operatorname{SL}_n$ on $m$-tuples of $n \times n$ matrices with entries in an
infinite field $K$. We show that invariants of degree $n^2- n$ define the null
cone. Consequently, invariants of degree $\leq n^6$ generate the ring of
invariants if $\operatorname{char}(K)=0$. We also prove that for $m \gg 0$,
invariants of degree at least $n\lfloor \sqrt{n+1}\rfloor$ are required to
define the null cone. We generalize our results to matrix invariants of
$m$-tuples of $p\times q$ matrices, and to rings of semi-invariants for
quivers. For the proofs, we use new techniques such as the regularity lemma by
Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor
blow-ups of matrix spaces. We will discuss several applications to algebraic
complexity theory, such as a deterministic polynomial time algorithm for
non-commutative rational identity testing, and the existence of small
division-free formulas for non-commutative polynomials.
