True statement from below …
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Let $A$ be $5times 5$ $textit{complex}$ matrix such that $(A^2-I)^2=0$. Assume that $A$ is not a diagonal matrix. Which of the following statement is true ?
$A$ is diagonalizable.
$A$ is NOT diagonalizable.
No conclusion can be drawn about the diagonalizability of $A$.
I feel that third option is correct but I don't know how to construct such counterexamples for first both options. Can someone please give me any hint ?
linear-algebra diagonalization
asked Nov 21 at 13:14
A. Bond
986
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up vote
0
down vote
favorite
Let $A$ be $5times 5$ $textit{complex}$ matrix such that $(A^2-I)^2=0$. Assume that $A$ is not a diagonal matrix. Which of the following statement is true ?
$A$ is diagonalizable.
$A$ is NOT diagonalizable.
No conclusion can be drawn about the diagonalizability of $A$.
I feel that third option is correct but I don't know how to construct such counterexamples for first both options. Can someone please give me any hint ?
linear-algebra diagonalization
asked Nov 21 at 13:14
A. Bond
986
You may consider examples of the form $A=pmatrix{B&0\ 0&I_3}$, where $B$ is $2times2$. Try to find a diagonal matrix $Bne I$ such that $(A^2-I)^2=0$. Then, try to find a non-diagonalisable $B$ such that $(A^2-I)^2=0$.
– user1551
Nov 21 at 13:17
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $A$ be $5times 5$ $textit{complex}$ matrix such that $(A^2-I)^2=0$. Assume that $A$ is not a diagonal matrix. Which of the following statement is true ?
$A$ is diagonalizable.
$A$ is NOT diagonalizable.
No conclusion can be drawn about the diagonalizability of $A$.
I feel that third option is correct but I don't know how to construct such counterexamples for first both options. Can someone please give me any hint ?
linear-algebra diagonalization
asked Nov 21 at 13:14
A. Bond
986
Let $A$ be $5times 5$ $textit{complex}$ matrix such that $(A^2-I)^2=0$. Assume that $A$ is not a diagonal matrix. Which of the following statement is true ?
$A$ is diagonalizable.
$A$ is NOT diagonalizable.
No conclusion can be drawn about the diagonalizability of $A$.
I feel that third option is correct but I don't know how to construct such counterexamples for first both options. Can someone please give me any hint ?
linear-algebra diagonalization
linear-algebra diagonalization
asked Nov 21 at 13:14
A. Bond
986
asked Nov 21 at 13:14
A. Bond
986
asked Nov 21 at 13:14
A. Bond
986
asked Nov 21 at 13:14
A. Bond
986
asked Nov 21 at 13:14
A. Bond
986
986
You may consider examples of the form $A=pmatrix{B&0\ 0&I_3}$, where $B$ is $2times2$. Try to find a diagonal matrix $Bne I$ such that $(A^2-I)^2=0$. Then, try to find a non-diagonalisable $B$ such that $(A^2-I)^2=0$.
– user1551
Nov 21 at 13:17
add a comment |
You may consider examples of the form $A=pmatrix{B&0\ 0&I_3}$, where $B$ is $2times2$. Try to find a diagonal matrix $Bne I$ such that $(A^2-I)^2=0$. Then, try to find a non-diagonalisable $B$ such that $(A^2-I)^2=0$.
– user1551
Nov 21 at 13:17
You may consider examples of the form $A=pmatrix{B&0\ 0&I_3}$, where $B$ is $2times2$. Try to find a diagonal matrix $Bne I$ such that $(A^2-I)^2=0$. Then, try to find a non-diagonalisable $B$ such that $(A^2-I)^2=0$.
– user1551
Nov 21 at 13:17
You may consider examples of the form $A=pmatrix{B&0\ 0&I_3}$, where $B$ is $2times2$. Try to find a diagonal matrix $Bne I$ such that $(A^2-I)^2=0$. Then, try to find a non-diagonalisable $B$ such that $(A^2-I)^2=0$.
– user1551
Nov 21 at 13:17
add a comment |
1 Answer
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The given condition is equivalent to say that $;A;$ is a zero of the polynomial $;(x^2-1)^2;$, so the minimal polynomial of $;A;,;;p_A(x);$ is a divisor of the above polynomial.
Remember now that $;A;$ is diagonalizable iff $;p_A(x);$ is a product of different linear polynomials, so since $;(x^2-1)^2=(x-1)^2(x+1)^2;$ , it is enough to find a matrix whose minimal polynomial is not $;(x-1),,,,(x+1),,,,text{or};;(x-1)(x+1);$ , for example
$$A=begin{pmatrix}
1&1&0&0&0\
0&1&0&0&0\
0&0&-1&0&0\
0&0&0&1&0\
0&0&0&0&1end{pmatrix}$$
is such a matrix.
answered Nov 21 at 13:39
DonAntonio
176k1491224
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The given condition is equivalent to say that $;A;$ is a zero of the polynomial $;(x^2-1)^2;$, so the minimal polynomial of $;A;,;;p_A(x);$ is a divisor of the above polynomial.
Remember now that $;A;$ is diagonalizable iff $;p_A(x);$ is a product of different linear polynomials, so since $;(x^2-1)^2=(x-1)^2(x+1)^2;$ , it is enough to find a matrix whose minimal polynomial is not $;(x-1),,,,(x+1),,,,text{or};;(x-1)(x+1);$ , for example
$$A=begin{pmatrix}
1&1&0&0&0\
0&1&0&0&0\
0&0&-1&0&0\
0&0&0&1&0\
0&0&0&0&1end{pmatrix}$$
is such a matrix.
answered Nov 21 at 13:39
DonAntonio
176k1491224
add a comment |
up vote
1
down vote
The given condition is equivalent to say that $;A;$ is a zero of the polynomial $;(x^2-1)^2;$, so the minimal polynomial of $;A;,;;p_A(x);$ is a divisor of the above polynomial.
Remember now that $;A;$ is diagonalizable iff $;p_A(x);$ is a product of different linear polynomials, so since $;(x^2-1)^2=(x-1)^2(x+1)^2;$ , it is enough to find a matrix whose minimal polynomial is not $;(x-1),,,,(x+1),,,,text{or};;(x-1)(x+1);$ , for example
$$A=begin{pmatrix}
1&1&0&0&0\
0&1&0&0&0\
0&0&-1&0&0\
0&0&0&1&0\
0&0&0&0&1end{pmatrix}$$
is such a matrix.
answered Nov 21 at 13:39
DonAntonio
176k1491224
add a comment |
up vote
1
down vote
up vote
1
down vote
The given condition is equivalent to say that $;A;$ is a zero of the polynomial $;(x^2-1)^2;$, so the minimal polynomial of $;A;,;;p_A(x);$ is a divisor of the above polynomial.
Remember now that $;A;$ is diagonalizable iff $;p_A(x);$ is a product of different linear polynomials, so since $;(x^2-1)^2=(x-1)^2(x+1)^2;$ , it is enough to find a matrix whose minimal polynomial is not $;(x-1),,,,(x+1),,,,text{or};;(x-1)(x+1);$ , for example
$$A=begin{pmatrix}
1&1&0&0&0\
0&1&0&0&0\
0&0&-1&0&0\
0&0&0&1&0\
0&0&0&0&1end{pmatrix}$$
is such a matrix.
answered Nov 21 at 13:39
DonAntonio
176k1491224
The given condition is equivalent to say that $;A;$ is a zero of the polynomial $;(x^2-1)^2;$, so the minimal polynomial of $;A;,;;p_A(x);$ is a divisor of the above polynomial.
Remember now that $;A;$ is diagonalizable iff $;p_A(x);$ is a product of different linear polynomials, so since $;(x^2-1)^2=(x-1)^2(x+1)^2;$ , it is enough to find a matrix whose minimal polynomial is not $;(x-1),,,,(x+1),,,,text{or};;(x-1)(x+1);$ , for example
$$A=begin{pmatrix}
1&1&0&0&0\
0&1&0&0&0\
0&0&-1&0&0\
0&0&0&1&0\
0&0&0&0&1end{pmatrix}$$
is such a matrix.
answered Nov 21 at 13:39
DonAntonio
176k1491224
answered Nov 21 at 13:39
DonAntonio
176k1491224
answered Nov 21 at 13:39
DonAntonio
176k1491224
answered Nov 21 at 13:39
DonAntonio
176k1491224
176k1491224
add a comment |
add a comment |
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You may consider examples of the form $A=pmatrix{B&0\ 0&I_3}$, where $B$ is $2times2$. Try to find a diagonal matrix $Bne I$ such that $(A^2-I)^2=0$. Then, try to find a non-diagonalisable $B$ such that $(A^2-I)^2=0$.
– user1551
Nov 21 at 13:17