G4 - Eigenspaces


Example 1

G4 - Eigenspaces (ver. 1)

The following matrix has an eigenvalue of 3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 2 & 1 & 0 & 1 \\ -2 & 4 & 0 & 3 \\ 5 & -4 & 3 & -6 \\ 1 & -2 & 0 & 3 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 2

G4 - Eigenspaces (ver. 2)

The following matrix has an eigenvalue of 2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 3 & 0 & -2 & 0 \\ 4 & 3 & -5 & 3 \\ 5 & 2 & -1 & 7 \\ 1 & -2 & -8 & -4 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ 0 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 3

G4 - Eigenspaces (ver. 3)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -1 & 6 & -3 & 6 \\ 3 & 2 & -3 & 6 \\ 1 & 2 & -5 & 2 \\ -3 & -6 & 3 & -10 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 4

G4 - Eigenspaces (ver. 4)

The following matrix has an eigenvalue of 2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 3 & -3 & 0 & -6 \\ 0 & 3 & -1 & 4 \\ -1 & 2 & 4 & -1 \\ -1 & 3 & 2 & 2 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 3 \\ -1 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 5

G4 - Eigenspaces (ver. 5)

The following matrix has an eigenvalue of -3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -2 & 1 & 0 & -2 \\ -1 & -3 & 2 & -5 \\ -4 & -4 & -2 & 5 \\ 1 & 1 & 0 & -5 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 1 \\ 1 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 6

G4 - Eigenspaces (ver. 6)

The following matrix has an eigenvalue of 4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 5 & 1 & 0 & 0 \\ 4 & 9 & 3 & 1 \\ 5 & 5 & 4 & 0 \\ 4 & 5 & 3 & 5 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 3 \\ -3 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 7

G4 - Eigenspaces (ver. 7)

The following matrix has an eigenvalue of 4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 6 & 0 & 4 & 2 \\ -1 & 7 & 4 & -4 \\ -1 & -1 & 0 & 0 \\ 1 & 2 & 6 & 3 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ -2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 8

G4 - Eigenspaces (ver. 8)

The following matrix has an eigenvalue of 4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 2 & -6 & 6 & -6 \\ -2 & -2 & -5 & -6 \\ -2 & -6 & 0 & -6 \\ 1 & 3 & -1 & 7 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -3 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -3 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 9

G4 - Eigenspaces (ver. 9)

The following matrix has an eigenvalue of 2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & -1 & 3 & 8 \\ 1 & -3 & 4 & 6 \\ 1 & -3 & 2 & 8 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 3 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 0 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 10

G4 - Eigenspaces (ver. 10)

The following matrix has an eigenvalue of 2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 3 & -4 & 0 & -2 \\ 2 & -6 & 1 & -2 \\ 2 & -8 & 1 & -6 \\ 2 & -8 & 4 & 6 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 4 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 0 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 11

G4 - Eigenspaces (ver. 11)

The following matrix has an eigenvalue of -2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 0 & 1 & -2 & 0 \\ -1 & -2 & 2 & -4 \\ 0 & 0 & -1 & -3 \\ -4 & -3 & 2 & 6 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 2 \\ 2 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 12

G4 - Eigenspaces (ver. 12)

The following matrix has an eigenvalue of -3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 4 & 4 & 3 & 4 \\ 2 & 2 & -3 & 5 \\ -3 & -3 & -3 & -3 \\ 3 & 1 & 2 & -2 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 13

G4 - Eigenspaces (ver. 13)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -3 & -4 & 1 & 5 \\ 2 & -11 & 2 & 8 \\ 0 & -4 & -3 & 5 \\ 0 & -4 & 2 & -2 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 0 \\ 2 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 14

G4 - Eigenspaces (ver. 14)

The following matrix has an eigenvalue of -1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -2 & -2 & -1 & 1 \\ 1 & 1 & 1 & -1 \\ 3 & 6 & 2 & -3 \\ -2 & -4 & -2 & 1 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 15

G4 - Eigenspaces (ver. 15)

The following matrix has an eigenvalue of 3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -1 & -3 & -3 & 0 \\ 0 & 4 & -2 & 6 \\ 5 & 4 & 6 & 2 \\ -2 & 0 & -1 & 5 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 0 \\ -2 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 16

G4 - Eigenspaces (ver. 16)

The following matrix has an eigenvalue of -3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -4 & -1 & 2 & -2 \\ -1 & -5 & 5 & -5 \\ -1 & -2 & 3 & -7 \\ 0 & -1 & 3 & -6 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ 3 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 17

G4 - Eigenspaces (ver. 17)

The following matrix has an eigenvalue of -2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -4 & -2 & 2 & 8 \\ 0 & -1 & -2 & -1 \\ 1 & 2 & -5 & -5 \\ 0 & -3 & 6 & 1 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ 2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 18

G4 - Eigenspaces (ver. 18)

The following matrix has an eigenvalue of 2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 3 & 2 & 1 & 3 \\ 0 & 3 & 2 & 2 \\ 2 & -1 & -6 & -4 \\ 1 & -2 & -7 & -3 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 3 \\ -2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ -2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 19

G4 - Eigenspaces (ver. 19)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -5 & 3 & 1 & 2 \\ 2 & -10 & -2 & -4 \\ 2 & -6 & -6 & -4 \\ -2 & 6 & 2 & 0 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 3 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 20

G4 - Eigenspaces (ver. 20)

The following matrix has an eigenvalue of -3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -2 & -2 & 4 & -3 \\ 0 & -2 & -3 & 2 \\ 0 & -2 & 3 & -4 \\ -1 & -1 & 5 & -6 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 2 \\ 3 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ -2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 21

G4 - Eigenspaces (ver. 21)

The following matrix has an eigenvalue of -1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -1 & 4 & -4 & 4 \\ 0 & 0 & -1 & 1 \\ -1 & 1 & 2 & -1 \\ -1 & 0 & 3 & -2 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 2 \\ 0 \\ 1 \\ 1 \end{array}\right]\right\}\)


Example 22

G4 - Eigenspaces (ver. 22)

The following matrix has an eigenvalue of -2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -1 & -2 & -7 & -4 \\ 1 & -3 & -3 & -1 \\ -1 & 1 & 2 & 1 \\ 0 & 2 & 5 & 4 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ -3 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 23

G4 - Eigenspaces (ver. 23)

The following matrix has an eigenvalue of -1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 6 & 3 & 1 & -7 \\ 2 & 0 & 0 & -1 \\ 0 & 1 & -2 & 4 \\ -1 & -1 & 0 & -1 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 24

G4 - Eigenspaces (ver. 24)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -3 & 0 & -5 & 5 \\ 2 & -3 & -5 & 3 \\ 1 & 0 & -8 & 4 \\ 0 & 0 & 5 & -9 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 0 \\ 2 \\ 1 \\ 1 \end{array}\right]\right\}\)


Example 25

G4 - Eigenspaces (ver. 25)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -5 & -3 & 3 & 2 \\ -1 & -7 & 3 & 2 \\ 2 & 6 & -10 & -4 \\ -2 & -6 & 6 & 0 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -3 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 26

G4 - Eigenspaces (ver. 26)

The following matrix has an eigenvalue of 3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 3 & -1 & 0 & -2 \\ 1 & 0 & -1 & -5 \\ 1 & -3 & 3 & -4 \\ -1 & 4 & -2 & 7 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ -2 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 27

G4 - Eigenspaces (ver. 27)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -3 & 2 & 0 & -2 \\ -3 & -10 & 0 & 6 \\ -1 & -2 & -4 & 2 \\ 3 & 6 & 0 & -10 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 28

G4 - Eigenspaces (ver. 28)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -6 & -8 & -6 & 4 \\ -1 & -8 & -3 & 2 \\ 1 & 4 & -1 & -2 \\ -2 & -8 & -6 & 0 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -4 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -3 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 29

G4 - Eigenspaces (ver. 29)

The following matrix has an eigenvalue of 3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 4 & 1 & -2 & -3 \\ -2 & 0 & -1 & -4 \\ -1 & -1 & 5 & 3 \\ 0 & 1 & 4 & 11 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ 0 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 30

G4 - Eigenspaces (ver. 30)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -5 & 0 & 1 & -1 \\ -1 & 2 & 0 & 8 \\ -2 & 5 & -3 & 6 \\ 0 & -2 & -2 & 0 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 2 \\ -1 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 31

G4 - Eigenspaces (ver. 31)

The following matrix has an eigenvalue of -3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -5 & 1 & 3 & 2 \\ -1 & -3 & 2 & 2 \\ 1 & 2 & -7 & -6 \\ 1 & -2 & 0 & -1 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 2 \\ 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 32

G4 - Eigenspaces (ver. 32)

The following matrix has an eigenvalue of 3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 6 & 6 & 0 & 6 \\ 1 & 5 & 0 & 2 \\ -1 & -2 & 3 & -2 \\ -2 & -4 & 0 & -1 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 33

G4 - Eigenspaces (ver. 33)

The following matrix has an eigenvalue of -2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -3 & -1 & -1 & 1 \\ -2 & -4 & -2 & 2 \\ 0 & 0 & -2 & 0 \\ 1 & 1 & 1 & -3 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 34

G4 - Eigenspaces (ver. 34)

The following matrix has an eigenvalue of 4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 6 & -1 & 3 & 8 \\ -1 & 5 & -2 & -5 \\ 0 & -3 & 8 & 7 \\ 1 & -1 & -3 & 4 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ 1 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 35

G4 - Eigenspaces (ver. 35)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -3 & -1 & 0 & -2 \\ 0 & -3 & 3 & 2 \\ -3 & 4 & -1 & 8 \\ -3 & 4 & 3 & 4 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -3 \\ -3 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ -2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 36

G4 - Eigenspaces (ver. 36)

The following matrix has an eigenvalue of 2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 1 & 1 & -2 & 4 \\ -1 & 2 & 2 & 3 \\ 0 & 1 & -2 & 1 \\ -1 & 0 & 2 & 5 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 2 \\ 4 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ -1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 37

G4 - Eigenspaces (ver. 37)

The following matrix has an eigenvalue of 3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 5 & 6 & 2 & 0 \\ 1 & 6 & 1 & 0 \\ -1 & -3 & 2 & 0 \\ 2 & 6 & 2 & 3 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -3 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 38

G4 - Eigenspaces (ver. 38)

The following matrix has an eigenvalue of -1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -2 & -1 & -1 & 3 \\ -1 & -1 & 0 & -3 \\ -4 & -4 & -4 & 8 \\ 1 & 1 & 1 & -4 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -3 \\ 2 \\ 4 \\ 1 \end{array}\right]\right\}\)


Example 39

G4 - Eigenspaces (ver. 39)

The following matrix has an eigenvalue of 1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 0 & 2 & 4 & 1 \\ -2 & 5 & 8 & 2 \\ -2 & 4 & 9 & 2 \\ -1 & 2 & 4 & 2 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 2 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 4 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 40

G4 - Eigenspaces (ver. 40)

The following matrix has an eigenvalue of -2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -1 & 0 & 3 & -5 \\ 0 & -1 & 2 & -6 \\ -1 & 1 & -2 & -3 \\ 0 & 1 & 3 & -10 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ 2 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 41

G4 - Eigenspaces (ver. 41)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -3 & 1 & 1 & 1 \\ -5 & -9 & -5 & -5 \\ 0 & 0 & -4 & 0 \\ 1 & 1 & 1 & -3 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 42

G4 - Eigenspaces (ver. 42)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -2 & 0 & 0 & 2 \\ 0 & -4 & 0 & 0 \\ -1 & 0 & -4 & -1 \\ 3 & 0 & 0 & -1 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 0 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 43

G4 - Eigenspaces (ver. 43)

The following matrix has an eigenvalue of -3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -5 & 1 & -1 & 2 \\ 3 & -5 & 0 & -2 \\ 0 & -2 & -9 & 4 \\ 2 & 0 & 4 & -7 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -2 \\ -3 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 44

G4 - Eigenspaces (ver. 44)

The following matrix has an eigenvalue of -1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -1 & -2 & -4 & 8 \\ 0 & 0 & 2 & -4 \\ 2 & 1 & -4 & 4 \\ -1 & -2 & -1 & 2 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 45

G4 - Eigenspaces (ver. 45)

The following matrix has an eigenvalue of 3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 4 & -2 & 4 & 7 \\ 0 & 4 & -2 & -4 \\ 2 & -2 & 8 & 7 \\ -1 & -1 & -2 & 4 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 1 \\ 2 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 46

G4 - Eigenspaces (ver. 46)

The following matrix has an eigenvalue of -4. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 0 & 3 & -4 & -6 \\ 5 & 2 & -5 & -3 \\ 3 & 5 & -7 & 1 \\ 3 & 2 & -3 & -9 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ -2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 47

G4 - Eigenspaces (ver. 47)

The following matrix has an eigenvalue of 2. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 0 & 1 & 0 & 1 \\ -1 & 2 & 1 & 3 \\ 0 & 0 & 3 & 2 \\ 0 & 1 & 0 & 1 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 1 \\ 1 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 48

G4 - Eigenspaces (ver. 48)

The following matrix has an eigenvalue of -3. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -2 & 4 & -1 & 1 \\ -1 & -7 & 2 & -3 \\ 0 & 0 & 1 & -8 \\ 2 & 8 & 2 & -9 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -4 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 49

G4 - Eigenspaces (ver. 49)

The following matrix has an eigenvalue of -1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} -1 & -1 & 0 & -1 \\ 1 & -5 & 4 & -7 \\ -1 & 4 & -4 & 6 \\ -1 & 1 & -7 & 6 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ 1 \end{array}\right]\right\}\)


Example 50

G4 - Eigenspaces (ver. 50)

The following matrix has an eigenvalue of 1. Explain how to find a basis for the eigenspace that corresponds to this eigenvalue.

\[\left[\begin{array}{rrrr} 2 & -1 & 2 & -1 \\ 0 & 1 & 0 & 0 \\ 2 & -2 & 5 & -2 \\ -1 & 1 & -2 & 2 \end{array}\right]\]

Answer.

\(\left\{\left[\begin{array}{r} 1 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)