We say that an matrix A is triangulizable if is similar to an upper triangular matrix B.
a. Give an example of a matrix with real entries that fails to be triangulizable over R .
b. Show that any matrix with complex entries is triangulizable over C . Hint: Give a proof by induction analogous to the proof of Theorem 8.1.1.
a) The triangulizable matrix over R since it complex eigen values are
we can conclude that matrix with complex entries is triangulizable over C
Triangularizable matrices are those that are similar to triangular matrices. In a nutshell, this is the same as stabilising a flag: upper triangular matrices are the ones that keep the standard flag, which is determined by the standard ordered basis.
Consider an A is triangulizable matrix is identical to and upper triangular matrix . If is an upper triangular matrix then the first column of S is an eigen vector of A. Hence any matrix without real eigen vectors fails to be triangulizable vector over R .Consider the matrix as
With real entries. To find the eigen values are as follows
The triangulizable matrix over R since it complex eigen values are
Consider the given problem for any matrix with complex entries is triangulizable over C .True value for (n-1) .A real eigen value of for A and eigen vector of length 1 for .For every A there exists a complex invertible matrix P whose first column is an eigen vector of A .It can be represented as below
Hypothesis of induction if B is triangulizable matrix such that there exists matrix Q that satisfies
Hence is upper triangular matrix
Hence, we can conclude that matrix with complex entries is triangulizable over C
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