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Deriving double angle trigonometric formulas using transformations. A really nice elementary proof.
In the previous post, two proofs were given for the Cauchy-Schwarz inequality. We will now consider another proof.
Definition 1 Let be an
matrix written as a
block matrix
where is a
matrix,
is a
matrix,
is a
, and
is a
, so
. Assuming
is nonsingular, then
is called the Schur complement of in
; or the Schur complement of
relative to
.
The Schur complement probably goes back to Carl Friedrich Gauss (1777-1855) (for Gaussian elimination). To solve the linear system
that is
by mimicking Gaussian elimination, that is, if is square and nonsingular, then by eliminating
, by multiplying the first equation by
and subtracting the second equation, we get
Note, the matrix is the Schur complement of
in
, and if it is square and nonsingular, then we can obtain the solution to our system.
The Schur complement comes up in Issai Schur’s (1875-1941) seminal lemma published in 1917, in which the Schur determinate formula was introduced. By considering elementary operations of partitioned matrices, let
where is square and nonsingular. We can change
so that the lower-left and upper-right submatrices become
. More precisely, we can make the lower-left and upper-right submatrices
by subtracting the first row multiplied by
from the second row, and by subtracting the first column multiplied by
from the second column. In symbols,
and in equation form,
Note that we have obtain the following factorization of :
By taking the determinants
we obtain the Schur’s determinant formula for block matrices,
Mathematician Emilie Virginia Haynsworth (1916-1985) introduced a name and a notation for the Schur complement of a square nonsingular (or invertible) submatrix in a partitioned (two-way block) matrix. The term Schur complement first appeared in Emily’s 1968 paper On the Schur Complement in Basel Mathematical Notes, then in Linear Algebra and its Applications Vol. 1 (1968), AMS Proceedings (1969), and in Linear Algebra and its Applications Vol. 3 (1970).
We will now present a block matrix proof, focusing on
complex matrices.
Proof: Let . Then
By taking the Schur complement of , we arrive at
and hence
which ensures, when and
are square, that
Equality occurs if and only if rank rank
; that is, by the Guttman rank additivity formula, rank
rank
rank
if and only if
. When
is nonsingular,
is nonsingular if and only if
is nonsingular.
References
[1] Zhang, Fuzhen. Matrix theory: basic results and techniques. Springer Science & Business Media, 2011.
http://www.phdcomics.com/comics/archive.php?comicid=27
Definition 1 A vector space over the number field
or
is called an inner product space if it is equipped with an inner product
satisfying for all
and scalar
,
,
if and only if
,
,
, and
.
is an inner product space over
with the inner product
An inner product space over is usually called a Euclidean space.
The following properties of an inner product can be deduced from the four axioms in Definition 1:
,
,
,
for all
if and only if
, and
.
An important property shared by all inner products is the Cauchy-Schwarz inequality and, for an inner product space, one of the most useful inequalities in mathematics.
Equality holds if and only if and
are linearly dependent.
The proof of this can be done in a number of different ways. The most common proof is to consider the quadratic function in
and derive the inequality from the non-positive discriminant. We will first present this proof.
Proof: Let be given. If
, the assertion is trivial, so we may assume that
. Let
and consider
which is a real quadratic polynomial with real coefficients. Because of axiom (1.), we know that for all real
, and hence
can have no real simple roots. The discriminant of
must therefore be non-positive
and hence
Since this inequality must hold for any pair of vectors, it must hold if is replaced by
, so we also have the inequality
But , so
If , then the statement of the theorem is trivial; if not, then we may divide equation (2) by the quantity
to obtain the desired inequality
Because of axiom (1.), can have a real (double) root only if
for some
. Thus, equality can occur in the discriminant condition in equation (1) if and only if
and
are linearly dependent.
We will now present a matrix proof, focusing on the complex vector space, which is perhaps the simplest proof of the Cauchy-Schwarz inequality.
Proof: For any vectors we noticed that,
By taking the determinant for the matrix,
the inequality follows at once,
Equality occurs if and only if the matrix
has rank 1; that is,
and
are linearly dependent.
References
[1] Zhang, Fuzhen. Matrix theory: basic results and techniques. Springer Science & Business Media, 2011.
Do you know these matrices described by Alan Rendall? If so, please point out a source where he may find more information about them. I am interested in knowing too!
I have come across a class of matrices with some interesting properties. I feel that they must be known but I have not been able to find anything written about them. This is probably just because I do not know the right place to look. I will describe these matrices here and I hope that somebody will be able to point out a source where I can find more information about them. Consider an $latex n\times n$ matrix $latex A$ with elements $latex a_{ij}$ having the following properties. The elements with $latex i=j$ (call them $latex b_i$) are negative. The elements with $latex j=i+1\ {\rm mod}\ n$ (call them $latex c_i$) are positive. All other elements are zero. The determinant of a matrix of this type is $latex \prod_i b_i+(-1)^{n+1}\prod_i c_i$. Notice that the two terms in this sum always have opposite signs. A property of these matrices which I…
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