Label the following statements as true or false. Assume that the characteristic polynomial of the matrix or linear operator splits. (a) The Jordan canonical form of a diagonal matrix is the matrix itself. (b) Let T be a linear operator on a finite-dimensional vector space V that has a Jordan canonical form J. If β is any basis for V, then the Jordan canonical form of [T]β is J. (c) Linear operators having the same characteristic polynomial are similar.(d) Matrices having the same Jordan canonical form are similar. (e) Every matrix is similar to its Jordan canonical form. (f) Every linear operator with the characteristic polynomial (−1)n (t −λ)n has the same Jordan canonical form. (g) Every linear operator on a finite-dimensional vector space has a unique Jordan canonical basis. (h) The dot diagrams of a linear operator on a finite-dimensional vector space are unique.
Label the following statements as true or false. Assume that the characteristic polynomial of the matrix or linear operator splits.
(a) The Jordan canonical form of a diagonal matrix is the matrix itself.
(b) Let T be a linear operator on a finite-dimensional vector space V that has a Jordan canonical form J. If β is any basis for V, then the Jordan canonical form of [T]β is J.
(c) Linear operators having the same characteristic polynomial are similar.(d) Matrices having the same Jordan canonical form are similar.
(e) Every matrix is similar to its Jordan canonical form.
(f) Every linear operator with the characteristic polynomial (−1)n (t −λ)n has the same Jordan canonical form.
(g) Every linear operator on a finite-dimensional vector space has a unique Jordan canonical basis.
(h) The dot diagrams of a linear operator on a finite-dimensional vector space are unique.
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