In matrix multiplication, there is a kind of matrix that plays a special role, such as 1 in the multiplication of numbers. This matrix is called identity matrix. It is a square matrix, and the elements on the diagonal line (called the main diagonal line) from the upper left corner to the lower right corner are 1. Everything else is 0.
According to the characteristics of identity matrix, any matrix multiplied by identity matrix is equal to itself, and the uniqueness of identity matrix is also widely used in higher mathematics.
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The nature of identity matrix:
According to the definition of matrix multiplication, the important properties of identity matrix are:
The eigenvalues of identity matrix are all 1, and any vector is the eigenvector of identity matrix.
Because the product of eigenvalues is equal to the determinant, the determinant of identity matrix is 1. Because the sum of eigenvalues is equal to the number of traces, the trace of identity matrix is n.