State whether the following statements are true or false: Matrix C is diagonaliz

Question

State whether the following statements are true or false: Matrix C is diagonalizable if it is similar to a diagonal matrix B; that is, there exists an invertible matrix P and B = PCP^-1 The eigenvalues of the matrix A = [1 -2 0 0 3 4 0 0 -7], are 1, 3, and 7 A square complex matrix A is called Unitary if its conjugate transpose equal to the matrix. The inner product of a nonzero vector u with itself ((u, u)) is always a positive real number If u = (1, 2, -3, 4) and v = (-2, 1, 4, 3) then (u, v) = 0, where (, ) denotes the Euclidean inner product. The matrix A = [4 2 - i 2 2 - i 3 6i - 3 2 -3 - 6i 2] is Hermitian.

Explanation / Answer

1. Only a square matrix is diagonizable. Not any matrix. Answer is false

2. False

3. False. A complex square matrix U is unitary if its conjugate transpose U is also its inverse.

4. True. Based on  positive-definite condition.

5. True

6. False. It is not Hermitian.

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