This a math presentation in engineering Matrix

Maths presentation

1. content RANK OF MATRIX CAYLEY-HAMILTON THEOREM DIAGONALIZATION OF MATRIX Matrices Unit 1 Participants Mayank Hargude IT58 Mayank Patil IT59 Mohammad Aqdas IT61 Mayur Sawarbandhe IT60

2. RANK OF MATRIX The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. In simpler terms, it represents the dimension of the space spanned by the rows or columns. A matrix's rank is essential in various mathematical applications, such as solving linear systems of equations and understanding the properties of transformations. For example, in a 3x3 matrix, if two rows are linearly dependent, the rank would be 2. The same applies to columns. The rank provides insights into the matrix's structure and can be determined through various methods, such as row reduction or using the determinant of minors.

3. C A Y L E Y - H A M I L T O N T H E O R E M The Cayley-Hamilton theorem states that for any square matrix A, substituting the matrix into its own characteristic polynomial yields the zero matrix. In other words, p(A) = 0, where p( λ ) is the characteristic polynomial of (A).

4. DIAGONALIZATION OF MATRIX B A B = 1 If ‘A’ is square matrix of order n having n linearly independent Eigen vector, then a non singular matrix B can be found such that B -1 AB is a diagonal form . If A is a square of order 3, having 3 linearly independent Eigen Vectors corresponding to Eigen Value λ 1, λ 2 , λ 3 then [ λ ] λ 0 0 λ Where B is Eigen Vector Matrix for Eigen value λ 1 , λ 2, λ 3 and B is called Model Matrix .

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