rotation matrix

 rotation matrix

https://en.wikipedia.org/wiki/Rotation_matrix 

matrix exponential

 

For diagonal matrix with eigen values: 

quantum speed up on alignment or alignment sequence analysis

 

alignment is a score matrix, so matrix method might speed the best alignment? 

https://www.youtube.com/watch?v=p2hZL38tqAs

quantum solution on the maximum unique match might be a good starting point. 

https://en.wikipedia.org/wiki/Sequence_alignment#Maximal_unique_match

alignment - free sequence 

https://bioinformaticsreview.com/20170704/role-of-information-theory-chaos-theory-and-linear-algebra-and-statistics-in-the-development-of-alignment-free-sequence-analysis/

Square of Ajacency matrice give estimation of path

Quantum Walk and Graph search. 

Qin: sequence alignment can be conerted all-2-all adjacent matrix. So, sequence alignment then might become a quantum walk and graph search problem. 

geometric multiplicity == algebraic multiplicity-for-a-symmetric-matrix

 good post to prove that geometric and algebraic multiplicity are the same for a symmetric matrix.

https://math.stackexchange.com/questions/393149/geometric-multiplicity-algebraic-multiplicity-for-a-symmetric-matrix

Matrix Eigen value

spectral theorem

https://youtu.be/KCANLl8z6PI

visual explanation
https://youtu.be/PFDu9oVAE-g

algebraic multiplicity and geometric multiplicity

https://people.math.carleton.ca/~kcheung/math/notes/MATH1107/wk10/10_algebraic_and_geometric_multiplicities.html

The algebraic multiplicity of λ is the number of times λ is repeated as a root of the characteristic polynomial.

 Let A be an n × n matrix with eigenvalue λ. The geometric multiplicity of λ is the dimension of the eigenspace of λ.

In general, the algebraic multiplicity and geometric multiplicity of an eigenvalue can differ. However, the geometric multiplicity can never exceed the algebraic multiplicity.

It is a fact that summing up the algebraic multiplicities of all the eigenvalues of an n×n$n\times n$ matrix A$A$ gives exactly n$n$. If for every eigenvalue of A$A$, the geometric multiplicity equals the algebraic multiplicity, then 
$A$ is said to be diagonalizable. As we will see, it is relatively easy to compute powers of a diagonalizable matrix.

SVD

good and clear explanations
https://youtu.be/EfZsEFhHcNM

least square method, matrix view

Khan academy
https://youtu.be/MC7l96tW8V8

Matrix diagonalization

The approach of this proof might be useful in the matrix approach for network aging study.

orthogonal projection, orthogonality

orthogonal projection of vector y to u:
y^hat = \frac{y \dot u}{u \dot u } u


U has orthonormal columns if and only if U^T U = I

inner product of vectors, dot products, orthogonality

$u \dot v = u^T v$

$u \dot v = v \dot u $

length (or norm) of vector $v$ is the square root of its inner product. This can be seen from the v [a,b], whose length(norm) is sqrt(a^2 + b^2)

u \dot v = ||u|| ||v|| cos \theta
||u-v|| = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos\theta

Two vector $u$ and $v$ are orthogonal if and only if $u \dot v = 0$. 

U has orthonormal columns if and only if U^T U = I

orthogonal projection of vector y to u:
y^hat = \frac{y \dot u}{u \dot u } u

Orthogonal projection of a point y to W space with {u1, u2, ... up} basis can be found by orthogonal projections on each base vector, u1, u2, ..., u_p.