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**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.

**Theorem 1**(Cauchy-Schwarz Inequality) Let be an inner product space. Then for all vectors and in over the field or ,

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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Had to reblog this post! I love linear algebra and programming. I am very interested in learning more about REDUCE and REDLOG.