Some of the things I've learned every day since Oct 10, 2016
Category Archives: linear algebra
January 1, 2017Posted by on
The general linear group , where is a vector space, is the group of all automorphisms under the operation of composition of linear transformations.
When is over the field and is finite-dimensional with , this group is isomorphic to the group of invertible matrices with entries from under the operation of matrix multiplication. In this case, the group is often written as .
General linear groups are used in group representations. A group representation is a representation of a group as a general linear group. That is, it is an automorphism
where is the group being represented and is any vector space.
December 31, 2016Posted by on
In abstract algebra, a multilinear form is a mapping
where is a vector space over the field , such that each argument of is linear over with the other arguments held fixed. A special case of this is when and is a bilinear form.
December 25, 2016Posted by on
2 sets of points in -dimensional Euclidean space are linearly separable if and only if there exists a non-zero vector and a number such that
holds for every , and does not hold for every . Intuitively, this means that two sets of points in an -dimensional Euclidean space are linearly separable if there is an -dimensional plane that when inserted into the same space separates the two sets.
(This concept could probably be extended to spaces which share certain properties with , such as having a partial order, closure, etc., but gives the simplest example.)
December 23, 2016Posted by on
In linear algebra, a bilinear map is a function , where are vector spaces over a common field, which is linear in each of its 2 components when the other is held fixed. When for all , it is referred to as a symmetric bilinear map.
Examples of bilinear maps include matrix multiplication, the inner product, and the cross product.
December 6, 2016Posted by on
A complex matrix is said to be positive-definite iff for every column vector , is real and positive. This is equivalent to the condition that all the eigenvalues of are positive.
Similarly, there are variations of positive-definiteness with analogous conditions:
positive-semidefinite real and non-negative non-negative eigenvalues
negative-semidefinite real and non-positive non-positive eigenvalues
negative-definite real and negative negative eigenvalues
December 5, 2016Posted by on
In linear algebra, a unitary operator over a inner product space is one which satisfies
Thus it is a special kind of normal operator. The following conditions are equivalent to being unitary:
- preserves the inner product. That is, .
- is distance-preserving. That is, .
- is unitary.
- is invertible and .
- is a normal operator with eigenvalues on the complex unit circle.
and the following are additionally true of a unitary operator:
- is normal.
- The eigenspaces of are orthogonal.
- Every eigenvalue of has an absolute value of .
- for some unitary transformations , where is diagonal.
(When is over it is sometimes referred to as ‘orthogonal’ rather than ‘unitary’.)
November 27, 2016Posted by on
A linear operator is self-adjoint iff it is its own adjoint, i.e. iff
This is equivalent to the condition that the matrix of with respect to any orthonormal basis is Hermitian (the matrix is its own conjugate transpose).
In addition, if is self-adjoint, then there exists an orthonormal eigenbasis for such that the matrix representation of with respect to is a diagonal matrix with real entries.
November 26, 2016Posted by on
Since a stationary distribution of a finite Markov chain satisfies , where is the transition matrix of , it can be seen as an eigenvector of eigenvalue under the linear transformation by . Specifically, is the intersection of the eigenspace with the hyperplane formed by the constraint that .
(Here the vector space in question is , where is the number of states in .)
November 21, 2016Posted by on
In linear algebra, if is a linear operator over a finite vector space , then there exists a unique , the adjoint operator, such that
for all .
November 13, 2016Posted by on
An orthonormal subset of an inner product space is a subspace for which all vectors are mutually orthogonal () and are unit vectors ().