Diagonalising Matrix
Diagonalizing a matrix involves finding a similarity transformation that transforms the matrix into a diagonal matrix. This process is important in various areas of mathematics and physics, as it simplifies calculations and reveals important properties of the original matrix. Diagonalizing a matrix allows for easier analysis and manipulation of its elements.
3 Key excerpts on "Diagonalising Matrix"
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...a matrix whose principal diagonal elements are the non-zero number µ, and whose remaining elements are all zero. This is sometimes denoted diag[ µ ] (provided that the dimension is clear from the context). The identity matrix is the scalar matrix that results when µ = 1. Illustration of this concept is very easy and is left to the reader. Property 1 We have tr(S) = nµ. Property 2 We have S − 1 = 1 μ I n = [ 1 μ δ i j ] It is easy to establish these properties, so, again, the proofs are left as exercises. Property 2 for scalar matrices, which may be obtained by direct consideration of the product S −1 S, is a special case of Property 5 for matrix inverses; see Exercise 1.17. 1.5.6 Diagonal matrix Just as the scalar matrix may be viewed as a simple generalization of the identity matrix, so the diagonal matrix may be viewed as a further generalization of the scalar matrix. A diagonal matrix is a square matrix of the form D = [ d i δ ij ] n×n, which is sometimes also written as diag[ d i ], where the subscript i now indicates that the diagonal elements are not necessarily all equal. Thus all of the elements of a diagonal matrix are zero except those on the principal diagonal, which are arbitrary scalars. Once again, such a matrix is simple to visualize. Property 1 We have tr (D) = ∑ i = 1 n d i. Property 2 We have D − 1 = [ 1 d i δ i j ] = diag [ 1 d i ] provided that d i ≠ 0 for all i. The proofs of these properties are easy. For the latter one, it will suffice to note that the operation of matrix multiplication gives that D D − 1 = [ d i δ i j ] [ 1 d i δ i j ] = [ δ i j ] n × n = I n 1.50 Therefore, by the definition of an inverse and its uniqueness, the result is established. 1.5.7 Transpose of a matrix If A is a matrix of order m × n, then the transpose of A, denoted A Τ, is the n × m matrix formed by interchanging the rows and columns of A...
eBook - ePubStatistical Methods for the Social and Behavioural Sciences
A Model-Based Approach
- David B. Flora(Author)
- 2017(Publication Date)
- SAGE Publications Ltd(Publisher)
...For any matrix, = X. A matrix consisting of only one column (i.e., with dimension I × 1) is called a column vector. Similarly, a matrix consisting of only one row (i.e., with dimension 1 × J) is called a row vector. A square matrix of dimension J has J rows and J columns. The entries c jj (that is, c 11, c 22,…, c JJ) of a square matrix C comprise the main diagonal of the matrix. The trace of a square matrix C, denoted tr(C), equals the sum of its diagonal elements. A square matrix A is symmetric if A = A′, that is, if a ij = a ji for all i and j. All correlation matrices, which are often denoted R, are square, symmetric matrices with each diagonal element equal to 1. A covariance matrix, which is often denoted S, is a square, symmetric matrix with off-diagonal elements that are the covariances between variables; the diagonal elements of a covariance matrix are the variances of the variables. A square matrix is called a diagonal matrix if all elements equal 0 except those on the main. diagonal. An important type of diagonal matrix is an identity matrix I, in which each diagonal element equals 1. Another important type of matrix (which need not be square) is a zero matrix, 0, in which all elements equal zero. Elementary matrix algebra Matrix addition and subtraction The major strength, indeed the purpose, of matrix algebra is that it provides a framework for algebraically manipulating large amounts of numerical entities using small, concise matrix expressions. As we will see, a set of many equations expressed using scalar terms can often be reexpressed using only a few (or even only one) matrix equation. The cost of this efficiency is that matrix algebra adheres to a strict set of rules, some of which are immediately intuitive and some of which are not. Two matrices can be added only if they have the same dimension; then their sum is formed by adding the corresponding elements...
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...For example, A = [ 0 0 0 0 0 0 0 0 0 ]. Identity Matrix : This matrix has column vectors called unit vectors, such as, A = [ 1 0 0 1 ], A = [ 1 0 0 0 1 0 0 0 1 ], A = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ], and so on. A Diagonal Matrix : All elements off the diagonal of this matrix. are zeros. A = [ 1 0 0 1 ], A = [ 2 0 0 0 3 0 0 0 4 ], ans so on. A Scalar Matrix : This is a matrix whose elements on the diagonal are equal. A = [ 1 0 0 1 ], A = [ 3 0 0 0 3 0 0 0 3 ], ans so on. A Triangular Matrix : A matrix whose lower and upper elements are arranged in such a way that they resemble a. triangle. A = [ 3 1 0 2 ], A = [ 2 6 7 0 5 9 0 0 3 ], A = [ 4 0 0 3 2 0 6 5 7 ], ans so on. Idempotent Matrix : This is a matrix having the property that A 2 = A. For example, A 2 = A, if A = [ 2 / 3 1 / 3 2 / 3 1 / 3 ], then AA = A 2 = [ 2 / 3 1 / 3 2 / 3 1 / 3 ]. Another example would be the identity matrix I. If I = I = [ 1 0 0 1 ], then I 2 = [ 1 0 0 1 ]. A Symmetric Matrix : A matrix is said to be symmetric if A = A t. A = [ 8 2 1 2 3 4 1 4 5 ] and A t = [ 8 2 1 2 3 4 1 4 5 ]. A Skew-Symmetric Matrix : This is a matrix whose A is equal to. − A 1. A = [ 0 4 − 4 0 ] and − A t = [ 0 4 − 4 0 ]. An Orthogonal Matrix : This is a matrix having the property that AA t = A t A = I. The Determinants The determinant of matrix A is denoted by I A I. To obtain I A I one should understand the following concepts. Minor : The minor IM ij I of an element in a matrix is obtained by deleting the i th row and the j th column of that matrix. Example 1: Find IM ij I for A = [ 3 7 4 8 ], where ij represents the i th row and the j th column respectively. | M 11 | = 8, | M 12 | = 4, | M 21 | = 7, and | M 22 | = 3. Example 2: Find |M ij |. for A = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 23 ]. Solution: | M 11 | = | a 22 a 23 a 32 a 33 |, | M 12 | = | a 21 a 23 a 31 a 33 |, | M 13 | = | a 21 a 22 a 31 a 32 |, | M 21 | = | a 12 a...
