Notation:
Defn: The transpose, , of an matrix is the matrix whose entries are given by
Defn: rank of matrix , rank: # of linearly independent columns of . We have
rank | dimcolumn space of | |
dimrow space of | ||
rank |
If is then rank.
For now: all matrices square .
If there is a matrix such that then we call the inverse of . If exists it is unique and and we write . The matrix has an inverse if and only if rank.
Inverses have the following properties:
Again is . The determinant if a function on the set of matrices such that:
det | ||
det | ||
det |
Here are some properties of the determinant:
Defn: Two vectors and are orthogonal if .
Defn: The inner product or dot product of and is
Defn: and are orthogonal if .
Defn: The norm (or length) of is
is orthogonal if each column of has length 1 and is orthogonal to each other column of .
Suppose is an matrix. The function
If is and and such that
Therefore det.
Conversely: if singular then there is such that .
Fact: det is polynomial in of degree .
Each root is an eigenvalue.
General the roots could be multiple roots or complex valued.
Matrix is diagonalized by a non-singular matrix if is diagonal.
If so then so each column of is eigenvector of with the th column having eigenvalue .
Thus to be diagonalizable must have linearly independent eigenvectors.
If is symmetric then
Defn: A symmetric matrix is non-negative definite if for all . It is positive definite if in addition implies .
is non-negative definite iff all its eigenvalues are non-negative.
is positive definite iff all eigenvalues positive.
A non-negative definite matrix has a symmetric non-negative definite square root. If
Suppose vector subspace of , basis for . Given any there is a unique which is closest to ; minimizes
Note that and that
Choose to minimize: minimize second term.
Achieved by making .
Since can take
Summary: closest point in is
Notice that the matrix is idempotent:
Suppose matrix, , and . Make matrix by putting in 2 by 2 matrix:
We can work with partitioned matrices just like ordinary matrices always making sure that in products we never change the order of multiplication of things.
Note partitioning of and must match.
Addition: dimensions of and must be the same.
Multiplication formula must have as many columns as has rows, etc.
In general: need to make sense for each .
Works with more than a 2 by 2 partitioning.
Defn: block diagonal matrix: partitioned matrix for which if . If
Partitioned inverses. Suppose , are symmetric positive definite. Look for inverse of
Solve to get