what is a positive definite matrix? how we can recognise this matrix without doing any matrix operations?

Answer 1

A positive definite matrix is a matrix A where all inner products
are positive where x can be any vector. If A is symmetric (or Hermitian), it is enough to know that all the eigenvalues of A are positve. There are no other good tests of positive definiteness.

Answer 2

Matrices are often used to perform transformations on coordinates. Any number of dimensions may be used. Matrices representing transformations can be multiplied together combining many translation matricies into one. This means that matrices can save time when performing translations with many stages as multiple rotations, translations, enlargements, etc can be combined together in one matrix and therefore executed in a single matrix operation.

A matrix is a concise and useful way of uniquely representing and working with linear transformations. For every linear transformation there exists exactly one corresponding matrix, thus every matrix corresponds to exactly one linear transformation.
A matirx is an array of numbers which can have any width and height. Below is an example of a 4x2 matrix. Note that the height is specified first and then the width.



Matricies Addition
The addition of two or more matricies is simply a matter of adding the values at the same position in each of the matricies. This does however impose the resistriction that all of the matricies must be the same size.



Matricies Subtraction
This is very similar to addition, but items at the corresponding positions are subtracted from each other. Once again the matricies must be the same size.



Matricies Multiplication
Multiplication is more complicated than what has been presented so far, therefore I will present a few examples and also discuss the process' involved in more detail.



To find the value of a point at row i and column j you must multiply the contents of row i in the first matrix by column j in the second matix. The example below highlights the values needed to be multiplied together to achieve the shown part of the answer.



In certain situations maticies that are not square may be multipled. The restriction for this is that the rows in the first matrix must be the same length as the columns in the second column. For example a 2x4 and a 4x2 matrix may be multiplied. This restriction is important as it means that a 2x4 cannot be multiplied by another 2x4.



The row and column rule for working out the answer shown in the previous example is needed here also. The figures below shows how the first two parts of the answer would be calculated.

Answer 3

In the real numbers, a positive definite matrix A is one such that for all real-valued nonzero vectors x, Ax > 0. Unfortunately, like many matrix properties, there is no way to identify certain properties without actually performing a calculation.

However, there are a number of different calculations you can carry out to verify that a matrix is positive definite. Some are given above. Another is as follows: a matrix is positive definite if and only if the prinicpal minors are positive.

Yet another is as follows: if B is any matrix, the matrix C = (B-transpose * B) is positive definite.

Answer 4

In linear algebra, a positive-definite matrix is simply a Hermitian matrix. There is no way to recognize a matrix for positive definiteness without doing any matrix operations.

If matrix operations are permitted, there are several tests to determine positive definiteness. E.g. det(A) >0 or {eigenvalues of A} >0 are sufficient conditions for positive definiteness.