By calculating the quadratic, cubic and quartic powers at most, we can know the n power, which strictly proves that mathematical induction is needed.

Only when the number of columns in the first matrix A is equal to the number of rows in the other matrix B can the multiplication of two matrices be defined. If a is an m×n matrix and b is an n×p matrix, their product c is an m×p matrix.

The positive definiteness of a symmetric matrix is closely related to its eigenvalue. A matrix is positive definite if and only if its eigenvalues are all positive numbers.

Using eigenvalue and eigenvector

Write the matrix a as pbp- 1, where p is a reversible matrix, b is a diagonal matrix, and a n = Pb NP- 1.

For example:

Find the law by calculating a 2 and a 3, and prove it by induction.

If r(A)= 1, then a = α β specialization t, and a n = (β tα) (n-1) a.

Note: β t α = α genus T β = TR (α β t)

With diagonalization a = p- 1diagp.

A^n = P^- 1diag^nP