This theorem is easy to prove: we only have to calculate the determinant of a diagonal matrix by cofactors. Look at the following solved exercise in which we find the determinant of a diagonal matrix by multiplying the elements on its main diagonal: The determinant of a diagonal matrix is the product of the elements on the main diagonal. To calculate the power of a diagonal matrix we must raise each element of the diagonal to the exponent: To solve a multiplication or a matrix product of two diagonal matrices we just have to multiply the elements of the diagonals with each other. The addition (and subtraction) of two diagonal matrices is very simple: you just have to add (or subtract) the numbers of the diagonals. Addition and subtraction of diagonal matrices That is why they are so used in mathematics. One of the reasons that diagonal matrices are so important to linear algebra is the ease with which they allow you to perform calculations. See: formula for adjoint of a matrix Operations with diagonal matrices The adjoint (or adjugate) of a diagonal matrix is another diagonal matrix. A square matrix is diagonal if and only if it is triangular and normal.The eigenvalues of a diagonal matrix are the elements of its main diagonal.Similarly, the null matrix is also a diagonal matrix because all its elements that are not on the diagonal are zeros, although the numbers on the diagonal are 0.The identity matrix is a diagonal matrix:.A diagonal matrix is an upper and lower triangular matrix at the same time.Any diagonal matrix is also a symmetric matrix (see symmetric matrix definition).This type of matrix is usually written indicating the numbers on the diagonal: Once we know the meaning of diagonal matrix, we are going to see several examples of diagonal matrices tu fully understand the concept: The entries on the main diagonal may or may not be null. A diagonal matrix is a square matrix in which all the elements outside the main diagonal are zero (0).
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