# Type-I seesaw: order of magnitude of the eigenvalues of effective $M_\nu$

Consider a square matrix $C$ constructed out of two other square matrices $A$ and $B$ as $$C=-A^TB^{-1}A.$$ Suppose all the elements of $B$ are very large compared to those of $A$. In such a case, is it guaranteed that all the matrix elements (or all the eigenvalues) of $C$ will also be small?

The preamble above looks like a math question but there is some physics context to it. In the type-I seesaw, the expression for the effective light neutrino mass matrix is given by $$M_\nu=-m_D^TM_R^{-1}m_D$$ where $m_D$ is the Dirac mass and $M_R$ is Majorana mass for the right-handed electroweak singlet $N_R$.

It is often stated that since "the matrix elements of $m_D$ $\ll$ the matrix elements of $M_R$", the eigenvalues of $M_\nu$ (representing light neutrino masses) will also be small? Or is it one of those hand-waving argument by physicists?

Take A to be an orthogonal rotation, so it does not alter the eigenvalues of 1/B. Choose then, for enormous M and minuscule ε, $$B= \begin{pmatrix} M&M-\epsilon\\ M-\epsilon &M \end{pmatrix} \qquad \Longrightarrow\qquad B^{-1}=\frac{1}{2\epsilon M-\epsilon^2} \begin{pmatrix} M&\epsilon-M\\ \epsilon -M&M \end{pmatrix}.$$
{"ft_lang_label":"__label__en","ft_lang_prob":0.9322842,"math_prob":0.99978846,"size":847,"snap":"2020-34-2020-40","text_gpt3_token_len":228,"char_repetition_ratio":0.12692764,"word_repetition_ratio":0.0,"special_character_ratio":0.26328218,"punctuation_ratio":0.060606062,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99998903,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-06T10:32:24Z\",\"WARC-Record-ID\":\"<urn:uuid:362a97f4-b176-406b-b3ff-cf3fb67f7405>\",\"Content-Length\":\"149083\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c8230d32-45ba-4f97-8449-802d09b8230c>\",\"WARC-Concurrent-To\":\"<urn:uuid:ad21341f-4d2e-4366-a13f-4b0ee6d54493>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/391157/type-i-seesaw-order-of-magnitude-of-the-eigenvalues-of-effective-m-nu\",\"WARC-Payload-Digest\":\"sha1:PSR2WD6BMVQ5ECT2X4UMKLFOVZY5H3XD\",\"WARC-Block-Digest\":\"sha1:CSOYP3LIQGGTNNDGI34IOLQKVC4EHOVC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439736902.24_warc_CC-MAIN-20200806091418-20200806121418-00489.warc.gz\"}"}