Square types of matrices have diagonal elements. These are the elements of I × j, where the value of i is equal to the value of j. For example, the first element in the first row of the first column and the third element in the third row of the third column.
Diagonal Elements of a Matrix
An element aij of a matrix A = [aij] is a diagonal elements of matrix if i = j, such as when rows and column suffixes are equal. Thus, a11 , a22 , a33, a44, … so on are diagonal elements of the matrix A = [aij].
The main diagonal of a square matrix (sometimes called just the diagonal) is the set of terms aii for which the row and column indices are the same, so from the upper left element to the lower right. For example, for the following matrix, the diagonal consists of 1, 6, 11, and 16.
However, a diagonal matrix is a square matrix which only has entries on its main diagonal and every other entry is zero. For example, the most common diagonal matrix is the identity matrix. Here is a 3 x 3 identity matrix. [ 1 0 0 0 1 0 0 0 1 ] .
A square matrix in which every element except the principal diagonal elements is zero is called a Diagonal Matrix. A square matrix D = [dij]n x n will be called a diagonal matrix if dij = 0, whenever i is not equal to j. There are many types of matrices like the Identity matrix.
Diagonalization Theorem
If and only if A has n linearly independent eigenvectors, then the n×n matrix A is diagonalizable. A=CDC-1 for this example.
A diagonal element refers to an element in a square matrix that lies on the principal diagonal, where the row and column indices are equal.
A diagonal matrix is defined as a square matrix in which all off-diagonal entries are zero. (Note that a diagonal matrix is necessarily symmetric.) Entries on the main diagonal may or may not be zero. If all entries on the main diagonal are equal scalars, then the diagonal matrix is called a scalar matrix.
D = diag( v ) returns a square diagonal matrix with the elements of vector v on the main diagonal. D = diag( v , k ) places the elements of vector v on the k th diagonal. k=0 represents the main diagonal, k>0 is above the main diagonal, and k<0 is below the main diagonal.
Every diagonal matrix is a square matrix. Identity matrix, null matrix, and scalar matrix are examples of a diagonal matrix as each of them has its non-principal diagonal elements to be zeros.
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