*true or false questions - please, give me a reason with your work(\"proof\" or

Question

*true or false questions - please, give me a reason with your work("proof" or examples)

* I really appriciate your time and your work below

Answer "True" or "False" to the following. Give reasons for your answers. The eigenvalues of an upper triangular matrix T are its diagonal entries. The eigenvalues of a real symmetric matrix are real. A matrix is nonsingular if and only if all its eigenvalues are nonzero. The eigenvalues of an orthogonal matrix are all equal to 1. An orthogonal matrix is not necessarily invertible. A real symmetric or a complex Hermitian matrix can be always transformed into a diagonal matrix by similarity transformation. Two similar matrices have the same eigenvalues. If two matrices have the same eigenvalues, they must be similar. The product of two upper (lower) triangular matrices does not need to be an upper (lower) triangular matrix. ||I|| = 1 for any norm. The length of a vector is preserved by an orthogonal multiplication. If ||A||

Explanation / Answer

1) false - because eigen values of only diagonal matrix are its diagonal elements

2)true - eigen values are always real untill and unless matix has imaginary numbers

3)true- for a non-singular matrix product of eigen values !0 . it is possible only when all eigen values are non zero

4)

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