Today I Learned
Some of the things I've learned every day since Oct 10, 2016
Category Archives: linear algebra
81: General Linear Groups
January 1, 2017
Posted 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.
80: Multilinear Forms
December 31, 2016
Posted 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.
74: Linearly Separable Values (Euclidean)
December 25, 2016
Posted 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.)
72: Bilinear Maps
December 23, 2016
Posted 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.
57: Positive-Definite Matrices
December 6, 2016
Posted 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
56: Unitary Linear Operators
December 5, 2016
Posted 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’.)
48: Self-Adjoint Linear Operators
November 27, 2016
Posted 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.
47: Invariant Distributions as Eigenvectors
November 26, 2016
Posted 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 .)
42: Adjoint Operators
November 21, 2016
Posted 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 .
34: Orthonormal Subsets
November 13, 2016
Posted by on An orthonormal subset of an inner product space is a subspace for which all vectors are mutually orthogonal () and are unit vectors ().
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