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To be a little more precise, the assumption here is that $\| error \|_\infty \le \epsilon$ and it seems you want to bound $\| error' \|_\infty$. So from $error' = A^{-1} error$ it follows that $\| err …

I haven't thought carefully about your argument, but I agree that the corollary must be false. Suppose that $A$ is a normal $2 \times 2$ real matrix. The corollary claims that $A$ is orthogonally sim …

This is a borderline suggestion, both in terms of how "major" it is and timing (does 1931 count as "early" 20th century?), but there is the Gershgorin circle theorem.

This formulation, or something very close to it (I don't have the book with me) is in Axler's Linear Algebra Done Right.

Given a linear map $T:H\to H$ on an inner-product space $H$ and a subspace $K\subseteq H$, define the map $T_K = \pi_K T \pi_K^* :K \to K$, where $\pi_K:H\to K$ is the orthogonal projection. As an im …

Here's an answer if you make a further assumption on your metric space: if $M$ is of strictly negative type, then it has $n-1$ negative eigenvalues, according to Lemma 3.6 of this paper. This conditi …

I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary fiel …

If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is finite …

You cannot get an upper bound in general in terms of the spectral radius $\rho(A)$. Counterexample: if $$ A = \begin{bmatrix} x & 1 \\ -x^2 & -x \end{bmatrix} $$ then $\rho(A) = 0$ and $s(A) = 2|x|$. …

With additional assumptions you can get an infinite series expansion, using the fact that $$ (I+B)^{-1} = \sum_{k=0}^\infty (-1)^k B^k $$ whenever $B$ is a square matrix with spectral radius $\rho(B) …

I recently taught a new (to my department) course titled "Matrix Analysis". For various reasons that I won't go into here, I was dissatisfied with the textbook I (loosely) followed, and with every ot …

More generally, you can form tensor products $V_1\otimes \cdots \otimes V_k$ of an arbitrary number of vector spaces, and a tensor refers to an element of one of these spaces, not just the case $k=2$. …

The closest thing I know for induced norms is the Riesz–Thorin theorem. There are other Hölder-like inequalities for matrices, for example involving Schatten norms.