No 2x2 examples please. I'm talking about large matrices that would be used in supercomputers.

The general result from eigendecomposition of a matrix:becomes:where $O$ is an orthogonal matrix, and therefore has $O^{-1}=O^T$.

A good definition is that the sparse matrix has non-zero entries proportional the number of rows. Therefore this is Big O notation less than something that has $N^2$ non zero entries. Of course, this only makes sense when generalizing to larger and larger matrices, otherwise we could take the constant of proportionality very high for one specific matrix.

Of course, this only makes sense when generalizing to larger and larger matrices, otherwise we could take the constant of proportionality very high for one specific matrix.

A matrix that equals its transpose:

Can represent a symmetric bilinear form as shown at matrix representation of a symmetric bilinear form, or a quadratic form.

WTF is a skew? "Antisymmetric" is just such a better name! And it also appears in other definitions such as antisymmetric multilinear map.

One of the most beautiful things is how they paywall even public domain works. E.g. here: www.nature.com/articles/119558a0 was published in 1927, and is therefore in the public domain as of 2023. But it is of course just paywalled as usual throughout 2023. There is zero incentive for them to open anything up.

When we have a symmetric matrix, a change of basis keeps symmetry iff it is done by an orthogonal matrix, in which case:

A member of the underlying field of a vector space. E.g. in ${\mathbb{R}}^3$, the underlying field is $\mathbb{R}$, and a scalar is a member of $\mathbb{R}$, i.e. a real number.

TODO what is the point of them? Why not just sum over every index that appears twice, regardless of where it is, as mentioned at: www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf.

Vectors with the index on top such as $x^i$ are the "regular vectors", they are called covariant vectors.

Those in indices on bottom are called contravariant vectors.

It is possible to change between them by Raising and lowering indices.

The values are different only when the metric signature matrix is different from the identity matrix.

Given the function $\psi$:the operator can be written in Planck units as:often written without function arguments as:Note how this looks just like the Laplacian in Einstein notation, since the d'Alembert operator is just a generalization of the laplace operator to Minkowski space.

Directly modelled by group.

For continuous symmetries, see: Lie group.