COLUMN:
VECTOR
VECTOR
When computing this measure of x in bbb C^n, the only thing that matters is its absolutely largest component
What is the "infinity norm"?
A matrix is said to have this "unhealthy" property when ||A|| ||A^(-1)|| is large
What is "ill-conditioned"?
This bread-and-butter technique for solving Ax=b might have been invented in China as early as 150 BCE
What is "Gaussian elimination"?
If you want Ax=b to have a unique solution, the matrix better have one of these subspaces in its domain
What is a "trivial nullspace"?
A non-orthogonal pair x,y might still produce x^Ty = 0 if x,y are both this type of vector
What is "complex-valued"?
This basic procedure for A = QR isn't even numerically stable (it should probably get therapy!)
What is "Gram-Schmidt orthogonalization"?
Straighten up and fly "right": this decomposition gives you an orthonormal basis for R(A)
What is the "QR factorization"?
Hmm... this kind of matrix doesn't seem to do ANYTHING to the inner product of x and y:
(:Mx,My:) = (:x,y:)
What is a "unitary matrix"?
When it comes to eigenvalues, these elements of bbb C^(n xx n) know how to "keep it bbb R"
What are "Hermitian matrices"?
Checking if rank([a_1 cdots a_n]) = n is a good way to determine if the set of vectors {a_1,ldots,a_n} has this standout property
What is "linear independence"?
This term applies to algorithms which "give you exactly what you almost asked for"
What is "backwards stable"?
The SVD of A in bbb C^(m xx n) gives you two of these special matrices, as well as one matrix of this other type?
What are "unitary matrices and a diagonal matrix"?
The quantity ||A||_M *||x|| is a bound for ||Ax|| whenever ||*||_M is one of these functions
What is a "consistent matrix norm"?
Use the formula (u^**v)/(u^**u)u to compute this new vector, which can be though of as "forcing one vector to align with another"
What is the "projection of v onto u"?
These "polynomial points" are (famously) ill-conditioned as a function of their standard basis coefficients
What are "polynomial roots"?
The minimizer of min_x||Ax-b||_2^2 is unique when A has this this status
What is "full column rank"?
There's a lot of V-shapes in the equation for this important factorization:
AV =V Lambda
Beware! If these two values don't match, you're gonna have a hard time finding an eigenbasis for lambda
What are "geometric and algebraic multiplicity"?
If a stable algorithm isn't accurate enough on your machine, find a computer with an improved level of this exacting quantity
What is "floating-point precision"?
It's a bad idea in practice to set up and solve these "standard relations" for the least-squares problem
What are the "normal equations"?
Our friend Cholesky is confident that you can write A = LL^T as long as A is one of these matrices.
What is a "symmetric positive-definite matrix"?