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 not have one of these subspaces in its domain
What is a "non-trivial nullspace"?
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"?
This basic procedure for A = QR isn't even numerically stable (unless it gets an adjustment!)
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"?
"The length of the difference is absolutely at least as big as the absolute difference of lengths!" is the "backwards" version of this famous relation for vector norms
What is the "triangle inequality"?
These "polynomial points" are (famously) ill-conditioned as a function of their standard basis coefficients
What are "polynomial roots"?
The solution 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 counts don't match, you're gonna have a hard time finding an eigenbasis for lambda
What are "geometric and algebraic multiplicity"?
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"?
If a stable algorithm isn't accurate enough on your machine, find a computer with an improved level of this exacting quantity
What is "machine epsilon"?
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"?