Let Find [T]B [T]B and P where
Linear map $T:\mathbb C^3\to\mathbb C^3$ $$ T\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}x+2y-i z\\[4pt]2y+i z\\[4pt] i x - z\end{pmatrix}. $$ Bases $$ B=\Big\{b_1=\begin{pmatrix}0\\i\\0\end{pmatrix},\; b_2=\begin{pmatrix}i\\1\\-1\end{pmatrix},\; b_3=\begin{pmatrix}0\\0\\2\end{pmatrix}\Big\},\qquad B'=\Big\{b'_1=\begin{pmatrix}1\\-i\\1\end{pmatrix},\; b'_2=\begin{pmatrix}0\\0\\1\end{pmatrix},\; b'_3=\begin{pmatrix}1\\i\\0\end{pmatrix}\Big\}. $$ Form the matrices whose columns are the basis vectors (these convert coordinates in the basis to standard coordinates): $$ S=[\,b_1\ b_2\ b_3\,]= \begin{pmatrix} 0 & i & 0\\[4pt] i & 1 & 0\\[4pt] 0 & -1 & 2 \end{pmatrix},\qquad S'=[\,b'_1\ b'_2\ b'_3\,]= \begin{pmatrix} 1 & 0 & 1\\[4pt] - i & 0 & i\\[4pt] 1 & 1 & 0 \end{pmatrix}. $$ --- ## 1) Compute $[T]_B$ For each $b_j\in B$ compute $T(b_j)$ (in standard coordinates) and then express $T(b_j)$ as a linear combination of _________ ___ _______ __________ ___ ______ _______ ______ ___ _________ _____ ________.
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Let M and Tbe a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year 15% of T's population moves to M and 10% of M's population moves to T. What is the long term effect of on the population of M and T? Are they likely to stabilise?
Find the Jordan canonical form J for
Also, find a matrix P such that J = P-1BP.
Let Find [T]B [T]B and P where
Check which of the following matrices is positive definite and which is positive semi-definite:
Also, find the square root of the positive definite matrix.
Which of the following statements are true and which are false? Give reasons for your answer.
i) If Vis a finite dimensional vector space and is a diagonalisable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of T is diagonal.
ii) Up to similarity, there is a unique 3 x 3 matrix with minimal polynomial (x - 1)2(x - 2).
iii) If a is the eigenvalue of a matrix A with characteristic polynomial f(x), (x - )k | f(x) and
then the geometric multiplicity of a is at most k.
iv) If = 1, then Ak → ∞ as k → ∞ .
v) If N is nilpotent, en is also nilpotent.
vi) The sum of two normal matrices of the order n is normal.
vii) If P and Q are positive definite operators, P + Q is a positive definite operator.
viii) Generalised inverse of a n x n matrix need not be unique.
ix) All the entries of a positive definite matrix are non-negative.
x) The SVD of any 2 x 3 matrix is unique.
Can A be similar to A + I ? Give reasons for your answer.
Let
Find a unitary matrix U such that U* AU is upper triangular.