linearAlgebra.jl
This unit contains linear algebra functions useful in relation to the Riemannian geometry of the manifold of Symmetric Positive Definite (SPD) or Hermitian Positive Definite (HPD) matrices. In Julia those are Hermitian matrices, see typecasting matrices.
In general they take a matrix as input (some may take other arrays as input) and are divided in eight categories depending on what kind of functions thay are and what they give as output:
| Category | Output |
|---|---|
| 1. Utilities | - - - |
| 2. Matrix normalizations and approximations | matrix |
| 3. Boolean functions of matrices | matrix |
| 4. Scalar functions of matrices | scalar |
| 5. Diagonal functions of matrices | diagonal matrix |
| 6. Unitary functions of matrices | orthogonal/unitary matrix |
| 7. Matrix function of matrices | matrix |
| 8. Spectral decompositions of positive matrices | spectral function of input |
| 9. Decompositions involving triangular matrices | triangular matrix |
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Utilities
| Function | Description |
|---|---|
typeofMatrix, typeofMat | Return the type of the matrix argument |
typeofVector, typeofVec | Return the type of the matrix vector argument |
dim | length of the dimensions of matrices and vectors of matrices |
remove | Remove one or more elements from a vector or one or more |
| columns or rows from a matrix | |
isSquare | Return true if matrix arguement is square, false otherwise |
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Missing docstring for typeofMatrix. Check Documenter's build log for details.
Missing docstring for typeofVector. Check Documenter's build log for details.
Missing docstring for dim. Check Documenter's build log for details.
Missing docstring for remove. Check Documenter's build log for details.
Missing docstring for isSquare. Check Documenter's build log for details.
Matrix normalizations and approximations
| Function | Description |
|---|---|
det1 | Normalize the determinant |
tr1 | Normalize the trace |
nearestPosDef | Nearest Symmetric/Hermitian Positive Semi-definite matrix |
nearestOrthogonalnearestOrth | Nearest Orthogonal matrix |
normalizeCol! | Normalize one or more columns |
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Missing docstring for det1. Check Documenter's build log for details.
Missing docstring for tr1. Check Documenter's build log for details.
Missing docstring for nearestPosDef. Check Documenter's build log for details.
Missing docstring for nearestOrthogonal. Check Documenter's build log for details.
Missing docstring for normalizeCol!. Check Documenter's build log for details.
Boolean functions of matrices
| Function | Description |
|---|---|
ispos | Check whether a real vector or diagonal matrix are comprised of all positive elements |
Missing docstring for ispos. Check Documenter's build log for details.
Scalar functions of matrices
| Function | Description |
|---|---|
colProd | Sum of products of the elements in two columns |
sumOfSqr, ss | Sum of squares of all elements or of specified columns |
sumOfSqrDiag, ssd | Sum of squares of the diagonal elements |
colNorm | Eucliden norm of a column |
sumOfSqrTril, sst | Sum of squares of the lower triangle elements up to a given underdiagonal |
tr | Fast trace of the product of two Hermitian matrices |
quadraticForm, qf | Fast quadratic form |
fidelity | (Quantum) Fidelity of two positive matrices |
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Missing docstring for colProd. Check Documenter's build log for details.
Missing docstring for sumOfSqr. Check Documenter's build log for details.
Missing docstring for sumOfSqrDiag. Check Documenter's build log for details.
Missing docstring for colNorm. Check Documenter's build log for details.
Missing docstring for sumOfSqrTril. Check Documenter's build log for details.
Missing docstring for tr. Check Documenter's build log for details.
Missing docstring for quadraticForm. Check Documenter's build log for details.
Missing docstring for fidelity. Check Documenter's build log for details.
Diagonal functions of matrices
| Function | Description |
|---|---|
fDiag, 𝑓𝔻 | Elemen-wise functions of matrix diagonals |
DiagOfProd, dop | Diagonal of the product of two matrices |
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Missing docstring for fDiag. Check Documenter's build log for details.
Missing docstring for DiagOfProd. Check Documenter's build log for details.
Unitary functions of matrices
| Function | Description |
|---|---|
mgs | Modified Gram-Schmidt orthogonalization |
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Missing docstring for mgs. Check Documenter's build log for details.
Matrix function of matrices
| Function | Description |
|---|---|
fVec | General function for multi-threaded computation of means and sums of matrix vectors |
congruence, cong | Compute congruent transformations |
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Missing docstring for fVec. Check Documenter's build log for details.
Missing docstring for congruence. Check Documenter's build log for details.
Spectral decompositions of positive matrices
| Function | Description |
|---|---|
evd | Eigenvalue-Eigenvector decomposition of a matrix in $UΛU'=P$ form |
frf | Full-rank factorization of an Hermitian matrix |
invfrf | Inverse of the full-rank factorization of an Hermitian matrix (whitening) |
spectralFunctions | Mother function for creating spectral functions of eigenvalues |
pow | Power of a positive matrix for any number of exponents in one pass |
invsqrt | Principal square root inverse (whitening) of a positive matrix |
sqr | Square of a positive matrix |
powerIterations, powIter | Power method for estimating any number of eigenvectors and associated eigenvalues |
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Missing docstring for evd. Check Documenter's build log for details.
Missing docstring for frf. Check Documenter's build log for details.
Missing docstring for invfrf. Check Documenter's build log for details.
Missing docstring for spectralFunctions. Check Documenter's build log for details.
Missing docstring for pow. Check Documenter's build log for details.
Missing docstring for invsqrt. Check Documenter's build log for details.
Missing docstring for sqr. Check Documenter's build log for details.
Missing docstring for powerIterations. Check Documenter's build log for details.
Decompositions involving triangular matrices
| Function | Description |
|---|---|
choL | Lower triangular factor of Cholesky decomposition |
choInv | Lower triangular factor of Cholesky decomposition and its inverse in one pass |
choInv! | as choInv, but destroying the input matrix |
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Missing docstring for choL. Check Documenter's build log for details.
Missing docstring for choInv. Check Documenter's build log for details.
Missing docstring for choInv!. Check Documenter's build log for details.