Consider a symmetric matrix A. If the vector is in the image of A and is in the kernel of A, is necessarily orthogonal to ? Justify your answer.
The is orthogonal to .
A theorem we studied in this book states that
And we know that A is symmetric , which implies
Hence is orthogonal to .
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.
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